Theorem spcimegft 2903

Description: A closed version of spcimegf 2906. (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 2833	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		spcimgft.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	issetf 2829	. . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4		exim 1652	. . . 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 1726	. . 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)))

This proof depends on syntax axioms:   → wi 4  ∀wal 1400   = wceq 1402  Ⅎwnf 1513  ∃wex 1545   ∈ wcel 2209  Ⅎwnfc 2379  Vcvv 2821

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is used by:  spcegft  2904  spcimegf  2906  spcimedv  2911