Theorem spcgft 3549

Description: A closed version of spcgf 3591. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
spcimgfi1.1	⊢ Ⅎ𝑥𝜓
spcimgfi1.2	⊢ Ⅎ𝑥𝐴

Assertion

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

Proof of Theorem spcgft

Step	Hyp	Ref	Expression
1		biimp 215	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
2	1	imim2i 16	. . 3 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → (𝜑 → 𝜓)))
3	2	alimi 1811	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)))
4		spcimgfi1.1	. . 3 ⊢ Ⅎ𝑥𝜓
5		spcimgfi1.2	. . 3 ⊢ Ⅎ𝑥𝐴
6	4, 5	spcimgfi1 3547	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)))
7	3, 6	syl 17	1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108  Ⅎwnfc 2890

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

This theorem is referenced by:  spcgf  3591  rspct  3608