Theorem spcimegft 2851

Description: A closed version of spcimegf 2854. (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 2783	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		spcimgft.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	issetf 2779	. . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4		exim 1622	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜑)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜓 → 𝜑)))
5	3, 4	biimtrid 152	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜑)) → (𝐴 ∈ V → ∃𝑥(𝜓 → 𝜑)))
6		spcimgft.1	. . . 4 ⊢ Ⅎ𝑥𝜓
7	6	19.37-1 1697	. . 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 1371   = wceq 1373  Ⅎwnf 1483  ∃wex 1515   ∈ wcel 2176  Ⅎwnfc 2335  Vcvv 2772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774

This theorem is referenced by:  spcegft  2852  spcimegf  2854  spcimedv  2859