Theorem ralxpxfr2d 3298

Description: Transfer a universal quantifier between one variable with pair-like semantics and two. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Hypotheses

Ref	Expression
ralxpxfr2d.a	⊢ 𝐴 ∈ V
ralxpxfr2d.b	⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴))
ralxpxfr2d.c	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
ralxpxfr2d	⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 𝜒))

Distinct variable groups:   𝜑,𝑥,𝑧   𝜑,𝑦,𝑥   𝜓,𝑦   𝜓,𝑧   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦,𝑧)   𝐴(𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧)   𝐶(𝑦,𝑧)   𝐷(𝑦,𝑧)

Proof of Theorem ralxpxfr2d

Step	Hyp	Ref	Expression
1		df-ral 2901	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
2		ralxpxfr2d.b	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴))
3	2	imbi1d 330	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 → 𝜓) ↔ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓)))
4	3	albidv 1836	. . . 4 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜓) ↔ ∀𝑥(∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓)))
5	1, 4	syl5bb 271	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓)))
6		ralcom4 3197	. . . 4 ⊢ (∀𝑦 ∈ 𝐶 ∀𝑥∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓) ↔ ∀𝑥∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓))
7		ralcom4 3197	. . . . 5 ⊢ (∀𝑧 ∈ 𝐷 ∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ ∀𝑥∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓))
8	7	ralbii 2963	. . . 4 ⊢ (∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 ∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ ∀𝑦 ∈ 𝐶 ∀𝑥∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓))
9		r19.23v 3005	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓) ↔ (∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓))
10	9	ralbii 2963	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓) ↔ ∀𝑦 ∈ 𝐶 (∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓))
11		r19.23v 3005	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐶 (∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓) ↔ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓))
12	10, 11	bitr2i 264	. . . . 5 ⊢ ((∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓) ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓))
13	12	albii 1737	. . . 4 ⊢ (∀𝑥(∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓) ↔ ∀𝑥∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓))
14	6, 8, 13	3bitr4ri 292	. . 3 ⊢ (∀𝑥(∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓) ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 ∀𝑥(𝑥 = 𝐴 → 𝜓))
15	5, 14	syl6bb 275	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 ∀𝑥(𝑥 = 𝐴 → 𝜓)))
16		ralxpxfr2d.c	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
17	16	pm5.74da 719	. . . . 5 ⊢ (𝜑 → ((𝑥 = 𝐴 → 𝜓) ↔ (𝑥 = 𝐴 → 𝜒)))
18	17	albidv 1836	. . . 4 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜒)))
19		ralxpxfr2d.a	. . . . 5 ⊢ 𝐴 ∈ V
20		biidd 251	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜒 ↔ 𝜒))
21	19, 20	ceqsalv 3206	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜒) ↔ 𝜒)
22	18, 21	syl6bb 275	. . 3 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ 𝜒))
23	22	2ralbidv 2972	. 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 ∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 𝜒))
24	15, 23	bitrd 267	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 𝜒))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173

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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175

This theorem is referenced by:  ralxpmap  7793