Theorem rspct 3608

Description: A closed version of rspc 3610. (Contributed by Andrew Salmon, 6-Jun-2011.)

Hypothesis

Ref	Expression
rspct.1	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
rspct	⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem rspct

Step	Hyp	Ref	Expression
1		df-ral 3062	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑))
2		eleq1 2829	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
3	2	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ (𝜑 ↔ 𝜓)) → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
4		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ (𝜑 ↔ 𝜓)) → (𝜑 ↔ 𝜓))
5	3, 4	imbi12d 344	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ (𝜑 ↔ 𝜓)) → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓)))
6	5	ex 412	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜓) → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))))
7	6	a2i 14	. . . . . 6 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))))
8	7	alimi 1811	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))))
9		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
10		rspct.1	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
11	9, 10	nfim 1896	. . . . . 6 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 → 𝜓)
12		nfcv 2905	. . . . . 6 ⊢ Ⅎ𝑥𝐴
13	11, 12	spcgft 3549	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))) → (𝐴 ∈ 𝐵 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜑) → (𝐴 ∈ 𝐵 → 𝜓))))
14	8, 13	syl 17	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜑) → (𝐴 ∈ 𝐵 → 𝜓))))
15	1, 14	syl7bi 255	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓))))
16	15	com34 91	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))))
17	16	pm2.43d 53	1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108  ∀wral 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062

This theorem is referenced by:  rspcdf  3609