Theorem spcimegft 2858

Description: A closed version of spcimegf 2861. (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 2788	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		spcimgft.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	issetf 2784	. . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4		exim 1623	. . . 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 1698	. . 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 1484  ∃wex 1516   ∈ wcel 2178  Ⅎwnfc 2337  Vcvv 2776

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778

This theorem is referenced by:  spcegft  2859  spcimegf  2861  spcimedv  2866