Theorem spcimegft 2759

Description: A closed version of spcimegf 2762. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

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

Assertion

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

Proof of Theorem spcimegft

Step	Hyp	Ref	Expression
1		elex 2692	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		spcimgft.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	issetf 2688	. . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4		exim 1578	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜑)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜓 → 𝜑)))
5	3, 4	syl5bi 151	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜑)) → (𝐴 ∈ V → ∃𝑥(𝜓 → 𝜑)))
6		spcimgft.1	. . . 4 ⊢ Ⅎ𝑥𝜓
7	6	19.37-1 1652	. . 3 ⊢ (∃𝑥(𝜓 → 𝜑) → (𝜓 → ∃𝑥𝜑))
8	5, 7	syl6 33	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜑)) → (𝐴 ∈ V → (𝜓 → ∃𝑥𝜑)))
9	1, 8	syl5 32	1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜑)) → (𝐴 ∈ 𝐵 → (𝜓 → ∃𝑥𝜑)))

Syntax hints:   → wi 4  ∀wal 1329   = wceq 1331  Ⅎwnf 1436  ∃wex 1468   ∈ wcel 1480  Ⅎwnfc 2266  Vcvv 2681

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:  spcegft  2760  spcimegf  2762  spcimedv  2767