Theorem spcimgfi1 3506

Description: A closed version of spcimgf 3509. (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 3505	. . 3 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)))
4	3	ex 412	. 2 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) → (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))))
5	1, 2, 4	mp2an 693	1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786  df-cleq 2729  df-clel 2812  df-nfc 2886

This theorem is referenced by:  spcgft  3508  spcimgf  3509  ss2iundf  44009  spcdvw  50032