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Theorem vtoclegft 3573
Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3549.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.) (Proof shortened by Wolf Lammen, 26-Jan-2025.)
Assertion
Ref Expression
vtoclegft ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → 𝜑)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem vtoclegft
StepHypRef Expression
1 biidd 261 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜑))
21ax-gen 1795 . . . . 5 𝑥(𝑥 = 𝐴 → (𝜑𝜑))
3 ceqsalt 3504 . . . . 5 ((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜑)) ∧ 𝐴𝐵) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜑))
42, 3mp3an2 1447 . . . 4 ((Ⅎ𝑥𝜑𝐴𝐵) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜑))
54ancoms 457 . . 3 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜑))
65biimpd 228 . 2 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑) → (∀𝑥(𝑥 = 𝐴𝜑) → 𝜑))
763impia 1115 1 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1085  wal 1537   = wceq 1539  wnf 1783  wcel 2104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-12 2169
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1087  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-clel 2808
This theorem is referenced by:  vtoclefex  36518
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