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 3143	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑))
2		eleq1 2900	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
3	2	adantr 483	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ (𝜑 ↔ 𝜓)) → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
4		simpr 487	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ (𝜑 ↔ 𝜓)) → (𝜑 ↔ 𝜓))
5	3, 4	imbi12d 347	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ (𝜑 ↔ 𝜓)) → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓)))
6	5	ex 415	. . . . . . 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 1808	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))))
9		nfv 1911	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
10		rspct.1	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
11	9, 10	nfim 1893	. . . . . 6 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 → 𝜓)
12		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑥𝐴
13	11, 12	spcgft 3586	. . . . 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 257	. . 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 208   ∧ wa 398  ∀wal 1531   = wceq 1533  Ⅎwnf 1780   ∈ wcel 2110  ∀wral 3138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3496

This theorem is referenced by:  rspcdf  3609