Theorem spcegft 2760

Description: A closed version of spcegf 2764. (Contributed by Jim Kingdon, 22-Jun-2018.)

Hypotheses

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

Assertion

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

Proof of Theorem spcegft

Step	Hyp	Ref	Expression
1		bi2 129	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
2	1	imim2i 12	. . 3 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → (𝜓 → 𝜑)))
3	2	alimi 1431	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜑)))
4		spcimgft.1	. . 3 ⊢ Ⅎ𝑥𝜓
5		spcimgft.2	. . 3 ⊢ Ⅎ𝑥𝐴
6	4, 5	spcimegft 2759	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜑)) → (𝐴 ∈ 𝐵 → (𝜓 → ∃𝑥𝜑)))
7	3, 6	syl 14	1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (𝜓 → ∃𝑥𝜑)))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329   = wceq 1331  Ⅎwnf 1436  ∃wex 1468   ∈ wcel 1480  Ⅎwnfc 2266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683

This theorem is referenced by:  spcegf  2764