Theorem spcimgfi1 3514

Description: A closed version of spcimgf 3517. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 27-Jul-2025.)

Hypotheses

Ref	Expression
spcimgfi1.1	⊢ Ⅎ𝑥𝜓
spcimgfi1.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
spcimgfi1	⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)))

Proof of Theorem spcimgfi1

Step	Hyp	Ref	Expression
1		spcimgfi1.2	. 2 ⊢ Ⅎ𝑥𝐴
2		spcimgfi1.1	. 2 ⊢ Ⅎ𝑥𝜓
3		spcimgft 3513	. 2 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)))
4	1, 2, 3	mpanl12 712	1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)))

Syntax hints:   → wi 4  ∀wal 1557   = wceq 1559  Ⅎwnf 1802   ∈ wcel 2141  Ⅎwnfc 2908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1799  df-nf 1803  df-cleq 2753  df-clel 2836  df-nfc 2910

This theorem is referenced by:  spcgft  3516  spcimgf  3517  ss2iundf  44199  spcdvw  50264