Theorem tfisi 6950

Description: A transfinite induction scheme in "implicit" form where the induction is done on an object derived from the object of interest. (Contributed by Stefan O'Rear, 24-Aug-2015.)

Hypotheses

Ref	Expression
tfisi.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
tfisi.b	⊢ (𝜑 → 𝑇 ∈ On)
tfisi.c	⊢ ((𝜑 ∧ (𝑅 ∈ On ∧ 𝑅 ⊆ 𝑇) ∧ ∀𝑦(𝑆 ∈ 𝑅 → 𝜒)) → 𝜓)
tfisi.d	⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
tfisi.e	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))
tfisi.f	⊢ (𝑥 = 𝑦 → 𝑅 = 𝑆)
tfisi.g	⊢ (𝑥 = 𝐴 → 𝑅 = 𝑇)

Assertion

Ref	Expression
tfisi	⊢ (𝜑 → 𝜃)

Distinct variable groups:   𝑥,𝑦,𝑇   𝑦,𝑅   𝑥,𝑆   𝜒,𝑥   𝜑,𝑥,𝑦   𝜓,𝑦   𝑥,𝐴   𝜃,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑦)   𝐴(𝑦)   𝑅(𝑥)   𝑆(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem tfisi

Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3587	. 2 ⊢ 𝑇 ⊆ 𝑇
2		eqid 2610	. . . . 5 ⊢ 𝑇 = 𝑇
3		tfisi.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		tfisi.b	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ On)
5		eqeq2 2621	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝑅 = 𝑧 ↔ 𝑅 = 𝑤))
6		sseq1 3589	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → (𝑧 ⊆ 𝑇 ↔ 𝑤 ⊆ 𝑇))
7	6	anbi2d 736	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) ↔ (𝜑 ∧ 𝑤 ⊆ 𝑇)))
8	7	imbi1d 330	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓) ↔ ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜓)))
9	5, 8	imbi12d 333	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → ((𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)) ↔ (𝑅 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜓))))
10	9	albidv 1836	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → (∀𝑥(𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)) ↔ ∀𝑥(𝑅 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜓))))
11		tfisi.f	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝑅 = 𝑆)
12	11	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑅 = 𝑤 ↔ 𝑆 = 𝑤))
13		tfisi.d	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
14	13	imbi2d 329	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜓) ↔ ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒)))
15	12, 14	imbi12d 333	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑅 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜓)) ↔ (𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))))
16	15	cbvalv 2261	. . . . . . . . 9 ⊢ (∀𝑥(𝑅 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜓)) ↔ ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒)))
17	10, 16	syl6bb 275	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (∀𝑥(𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)) ↔ ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))))
18		eqeq2 2621	. . . . . . . . . 10 ⊢ (𝑧 = 𝑇 → (𝑅 = 𝑧 ↔ 𝑅 = 𝑇))
19		sseq1 3589	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑇 → (𝑧 ⊆ 𝑇 ↔ 𝑇 ⊆ 𝑇))
20	19	anbi2d 736	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑇 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) ↔ (𝜑 ∧ 𝑇 ⊆ 𝑇)))
21	20	imbi1d 330	. . . . . . . . . 10 ⊢ (𝑧 = 𝑇 → (((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓) ↔ ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓)))
22	18, 21	imbi12d 333	. . . . . . . . 9 ⊢ (𝑧 = 𝑇 → ((𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)) ↔ (𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓))))
23	22	albidv 1836	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (∀𝑥(𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)) ↔ ∀𝑥(𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓))))
24		simp3l 1082	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝜑)
25		simp2 1055	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝑅 = 𝑧)
26		simp1l 1078	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝑧 ∈ On)
27	25, 26	eqeltrd 2688	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝑅 ∈ On)
28		simp3r 1083	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝑧 ⊆ 𝑇)
29	25, 28	eqsstrd 3602	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝑅 ⊆ 𝑇)
30		simpl3l 1109	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → 𝜑)
31		simpl1l 1105	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → 𝑧 ∈ On)
32		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅)
33		simpl2 1058	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → 𝑅 = 𝑧)
34	32, 33	eleqtrd 2690	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑧)
35		onelss 5683	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ On → (⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑧 → ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑧))
36	31, 34, 35	sylc 63	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑧)
37		simpl3r 1110	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → 𝑧 ⊆ 𝑇)
38	36, 37	sstrd 3578	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇)
39		simpl1r 1106	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒)))
40		eqeq2 2621	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → (𝑆 = 𝑤 ↔ 𝑆 = ⦋𝑣 / 𝑥⦌𝑅))
41		sseq1 3589	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → (𝑤 ⊆ 𝑇 ↔ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇))
42	41	anbi2d 736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) ↔ (𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇)))
43	42	imbi1d 330	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → (((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒) ↔ ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒)))
44	40, 43	imbi12d 333	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒)) ↔ (𝑆 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒))))
45	44	albidv 1836	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = ⦋𝑣 / 𝑥⦌𝑅 → (∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒)) ↔ ∀𝑦(𝑆 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒))))
46	45	rspcva 3280	. . . . . . . . . . . . . . . . . 18 ⊢ ((⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑧 ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) → ∀𝑦(𝑆 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒)))
47	34, 39, 46	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → ∀𝑦(𝑆 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒)))
48		eqidd 2611	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → ⦋𝑣 / 𝑥⦌𝑅 = ⦋𝑣 / 𝑥⦌𝑅)
49		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑥𝑦
50		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑥𝑆
51	49, 50, 11	csbhypf 3518	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = 𝑦 → ⦋𝑣 / 𝑥⦌𝑅 = 𝑆)
52	51	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = 𝑦 → 𝑆 = ⦋𝑣 / 𝑥⦌𝑅)
53	52	equcoms 1934	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑣 → 𝑆 = ⦋𝑣 / 𝑥⦌𝑅)
54	53	eqeq1d 2612	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑣 → (𝑆 = ⦋𝑣 / 𝑥⦌𝑅 ↔ ⦋𝑣 / 𝑥⦌𝑅 = ⦋𝑣 / 𝑥⦌𝑅))
55		nfv 1830	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑥𝜒
56	55, 13	sbhypf 3226	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = 𝑦 → ([𝑣 / 𝑥]𝜓 ↔ 𝜒))
57	56	bicomd 212	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = 𝑦 → (𝜒 ↔ [𝑣 / 𝑥]𝜓))
58	57	equcoms 1934	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑣 → (𝜒 ↔ [𝑣 / 𝑥]𝜓))
59	58	imbi2d 329	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑣 → (((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒) ↔ ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → [𝑣 / 𝑥]𝜓)))
60	54, 59	imbi12d 333	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑣 → ((𝑆 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒)) ↔ (⦋𝑣 / 𝑥⦌𝑅 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → [𝑣 / 𝑥]𝜓))))
61	60	spv 2248	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦(𝑆 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → 𝜒)) → (⦋𝑣 / 𝑥⦌𝑅 = ⦋𝑣 / 𝑥⦌𝑅 → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → [𝑣 / 𝑥]𝜓)))
62	47, 48, 61	sylc 63	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → ((𝜑 ∧ ⦋𝑣 / 𝑥⦌𝑅 ⊆ 𝑇) → [𝑣 / 𝑥]𝜓))
63	30, 38, 62	mp2and 711	. . . . . . . . . . . . . . 15 ⊢ ((((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) ∧ ⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅) → [𝑣 / 𝑥]𝜓)
64	63	ex 449	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → (⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅 → [𝑣 / 𝑥]𝜓))
65	64	alrimiv 1842	. . . . . . . . . . . . 13 ⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → ∀𝑣(⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅 → [𝑣 / 𝑥]𝜓))
66	51	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑦 → (⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅 ↔ 𝑆 ∈ 𝑅))
67	66, 56	imbi12d 333	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑦 → ((⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅 → [𝑣 / 𝑥]𝜓) ↔ (𝑆 ∈ 𝑅 → 𝜒)))
68	67	cbvalv 2261	. . . . . . . . . . . . 13 ⊢ (∀𝑣(⦋𝑣 / 𝑥⦌𝑅 ∈ 𝑅 → [𝑣 / 𝑥]𝜓) ↔ ∀𝑦(𝑆 ∈ 𝑅 → 𝜒))
69	65, 68	sylib 207	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → ∀𝑦(𝑆 ∈ 𝑅 → 𝜒))
70		tfisi.c	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑅 ∈ On ∧ 𝑅 ⊆ 𝑇) ∧ ∀𝑦(𝑆 ∈ 𝑅 → 𝜒)) → 𝜓)
71	24, 27, 29, 69, 70	syl121anc 1323	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) ∧ 𝑅 = 𝑧 ∧ (𝜑 ∧ 𝑧 ⊆ 𝑇)) → 𝜓)
72	71	3exp 1256	. . . . . . . . . 10 ⊢ ((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) → (𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)))
73	72	alrimiv 1842	. . . . . . . . 9 ⊢ ((𝑧 ∈ On ∧ ∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒))) → ∀𝑥(𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓)))
74	73	ex 449	. . . . . . . 8 ⊢ (𝑧 ∈ On → (∀𝑤 ∈ 𝑧 ∀𝑦(𝑆 = 𝑤 → ((𝜑 ∧ 𝑤 ⊆ 𝑇) → 𝜒)) → ∀𝑥(𝑅 = 𝑧 → ((𝜑 ∧ 𝑧 ⊆ 𝑇) → 𝜓))))
75	17, 23, 74	tfis3 6949	. . . . . . 7 ⊢ (𝑇 ∈ On → ∀𝑥(𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓)))
76	4, 75	syl 17	. . . . . 6 ⊢ (𝜑 → ∀𝑥(𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓)))
77		tfisi.g	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → 𝑅 = 𝑇)
78	77	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑅 = 𝑇 ↔ 𝑇 = 𝑇))
79		tfisi.e	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))
80	79	imbi2d 329	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓) ↔ ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜃)))
81	78, 80	imbi12d 333	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓)) ↔ (𝑇 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜃))))
82	81	spcgv 3266	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑅 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜓)) → (𝑇 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜃))))
83	3, 76, 82	sylc 63	. . . . 5 ⊢ (𝜑 → (𝑇 = 𝑇 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜃)))
84	2, 83	mpi 20	. . . 4 ⊢ (𝜑 → ((𝜑 ∧ 𝑇 ⊆ 𝑇) → 𝜃))
85	84	expd 451	. . 3 ⊢ (𝜑 → (𝜑 → (𝑇 ⊆ 𝑇 → 𝜃)))
86	85	pm2.43i 50	. 2 ⊢ (𝜑 → (𝑇 ⊆ 𝑇 → 𝜃))
87	1, 86	mpi 20	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  [wsb 1867   ∈ wcel 1977  ∀wral 2896  ⦋csb 3499   ⊆ wss 3540  Oncon0 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644

This theorem is referenced by:  indcardi  8747