Theorem spcimegft 2838

Description: A closed version of spcimegf 2841. (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 2771	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		spcimgft.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	issetf 2767	. . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4		exim 1610	. . . 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 1685	. . 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 1362   = wceq 1364  Ⅎwnf 1471  ∃wex 1503   ∈ wcel 2164  Ⅎwnfc 2323  Vcvv 2760

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762

This theorem is referenced by:  spcegft  2839  spcimegf  2841  spcimedv  2846