MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtoclegft Structured version   Visualization version   GIF version

Theorem vtoclegft 3544
Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 3517.) (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 262 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜑))
21ax-gen 1798 . . . . 5 𝑥(𝑥 = 𝐴 → (𝜑𝜑))
3 ceqsalt 3477 . . . . 5 ((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜑)) ∧ 𝐴𝐵) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜑))
42, 3mp3an2 1450 . . . 4 ((Ⅎ𝑥𝜑𝐴𝐵) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜑))
54ancoms 460 . . 3 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜑))
65biimpd 228 . 2 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑) → (∀𝑥(𝑥 = 𝐴𝜑) → 𝜑))
763impia 1118 1 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088  wal 1540   = wceq 1542  wnf 1786  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-12 2172
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-clel 2811
This theorem is referenced by:  vtoclefex  35855
  Copyright terms: Public domain W3C validator