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Theorem spcimgft 3546
Description: Closed theorem form of spcimgf 3550. (Contributed by Wolf Lammen, 28-Jul-2025.)
Assertion
Ref Expression
spcimgft (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝐴𝑉 → (∀𝑥𝜑𝜓)))

Proof of Theorem spcimgft
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elissetv 2820 . . . 4 (𝐴𝑉 → ∃𝑦 𝑦 = 𝐴)
2 cbvexeqsetf 3493 . . . 4 (𝑥𝐴 → (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴))
31, 2imbitrrid 246 . . 3 (𝑥𝐴 → (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴))
4 pm2.04 90 . . . . . 6 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝜑 → (𝑥 = 𝐴𝜓)))
54al2imi 1812 . . . . 5 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝐴𝜓)))
6 19.23t 2208 . . . . . 6 (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴𝜓) ↔ (∃𝑥 𝑥 = 𝐴𝜓)))
76biimpd 229 . . . . 5 (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴𝜓) → (∃𝑥 𝑥 = 𝐴𝜓)))
85, 7sylan9r 508 . . . 4 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (∀𝑥𝜑 → (∃𝑥 𝑥 = 𝐴𝜓)))
98com23 86 . . 3 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (∃𝑥 𝑥 = 𝐴 → (∀𝑥𝜑𝜓)))
103, 9sylan9 507 . 2 ((𝑥𝐴 ∧ (Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)))) → (𝐴𝑉 → (∀𝑥𝜑𝜓)))
1110anassrs 467 1 (((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → (𝐴𝑉 → (∀𝑥𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wex 1776  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:  spcimgfi1  3547  vtoclgft  3552
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