Theorem bnj1468 30170

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1468.1	⊢ (𝜓 → ∀𝑥𝜓)
bnj1468.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
bnj1468.3	⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Assertion

Ref	Expression
bnj1468	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

Distinct variable groups:   𝑦,𝐴   𝑦,𝑉   𝜑,𝑦   𝜓,𝑦   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem bnj1468

Step	Hyp	Ref	Expression
1		sbcco 3425	. 2 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)
2		ax-5 1827	. . 3 ⊢ (𝜓 → ∀𝑦𝜓)
3		bnj1468.3	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
4	3	nfcii 2742	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
5	4	nfeq2 2766	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐴
6		nfsbc1v 3422	. . . . . . 7 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
7		bnj1468.1	. . . . . . . 8 ⊢ (𝜓 → ∀𝑥𝜓)
8	7	nf5i 2011	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
9	6, 8	nfbi 1821	. . . . . 6 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 ↔ 𝜓)
10	5, 9	nfim 1813	. . . . 5 ⊢ Ⅎ𝑥(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
11	10	nf5ri 2053	. . . 4 ⊢ ((𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) → ∀𝑥(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)))
12		ax6ev 1877	. . . . 5 ⊢ ∃𝑥 𝑥 = 𝑦
13		eqeq1 2614	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
14		bnj1468.2	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
15	13, 14	syl6bir 243	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝐴 → (𝜑 ↔ 𝜓)))
16		sbceq1a 3413	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
17	16	bibi1d 332	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)))
18	15, 17	sylibd 228	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)))
19	12, 18	bnj101 30043	. . . 4 ⊢ ∃𝑥(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
20	11, 19	bnj1131 30112	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))
21	2, 20	bnj1464 30168	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜓))
22	1, 21	syl5bbr 273	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  [wsbc 3402

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-v 3175  df-sbc 3403

This theorem is referenced by:  bnj1463  30377