Theorem rspct 3562

Description: A closed version of rspc 3564. (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 3077	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑))
2		eleq1 2848	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
3	2	adantr 486	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ (𝜑 ↔ 𝜓)) → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
4		simpr 490	. . . . . . . . 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 418	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜓) → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))))
7	6	a2i 15	. . . . . 6 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))))
8	7	alimi 1843	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))))
9		nfv 1946	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
10		rspct.1	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
11	9, 10	nfim 1928	. . . . . 6 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 → 𝜓)
12		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑥𝐴
13	11, 12	spcgft 3512	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))) → (𝐴 ∈ 𝐵 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜑) → (𝐴 ∈ 𝐵 → 𝜓))))
14	8, 13	syl 18	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜑) → (𝐴 ∈ 𝐵 → 𝜓))))
15	1, 14	syl7bi 258	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → (𝐴 ∈ 𝐵 → 𝜓))))
16	15	com34 92	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓))))
17	16	pm2.43d 54	1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2144  ∀wral 3076

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077

This theorem is used by:  rspcdf  3563