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Theorem spcimgfi1 3547
Description: A closed version of spcimgf 3550. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 27-Jul-2025.)
Hypotheses
Ref Expression
spcimgfi1.1 𝑥𝜓
spcimgfi1.2 𝑥𝐴
Assertion
Ref Expression
spcimgfi1 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝜑𝜓)))

Proof of Theorem spcimgfi1
StepHypRef Expression
1 spcimgfi1.2 . 2 𝑥𝐴
2 spcimgfi1.1 . 2 𝑥𝜓
3 spcimgft 3546 . . 3 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝐴𝐵 → (∀𝑥𝜑𝜓)))
43ex 412 . 2 ((𝑥𝐴 ∧ Ⅎ𝑥𝜓) → (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝜑𝜓))))
51, 2, 4mp2an 692 1 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wnf 1780  wcel 2106  wnfc 2888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-cleq 2727  df-clel 2814  df-nfc 2890
This theorem is referenced by:  spcgft  3549  spcimgf  3550  ss2iundf  43649  spcdvw  48910
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