Theorem rspct 3275

Description: A closed version of rspc 3276. (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 2901	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑))
2		eleq1 2676	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
3	2	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ (𝜑 ↔ 𝜓)) → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
4		simpr 476	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ (𝜑 ↔ 𝜓)) → (𝜑 ↔ 𝜓))
5	3, 4	imbi12d 333	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ (𝜑 ↔ 𝜓)) → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓)))
6	5	ex 449	. . . . . . 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 1730	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))))
9		nfv 1830	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
10		rspct.1	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
11	9, 10	nfim 1813	. . . . . 6 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 → 𝜓)
12		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑥𝐴
13	11, 12	spcgft 3258	. . . . 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 244	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓))))
16	15	com34 89	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))))
17	16	pm2.43d 51	1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  ∀wral 2896

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-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-v 3175

This theorem is referenced by:  rspcdf  42222