Theorem spcimgfi1 3515

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 3514	. 2 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)))
4	1, 2, 3	mpanl12 715	1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2147  Ⅎwnfc 2911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-cleq 2755  df-clel 2839  df-nfc 2913

This theorem is used by:  spcgft  3516  spcimgf  3517  spcdvw  10000  ss2iundf  44862