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

Theorem ceqsal 3518
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) Avoid df-clab 2714. (Revised by Wolf Lammen, 23-Jan-2025.)
Hypotheses
Ref Expression
ceqsal.1 𝑥𝜓
ceqsal.2 𝐴 ∈ V
ceqsal.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsal (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem ceqsal
StepHypRef Expression
1 ceqsal.1 . . 3 𝑥𝜓
2119.23 2210 . 2 (∀𝑥(𝑥 = 𝐴𝜓) ↔ (∃𝑥 𝑥 = 𝐴𝜓))
3 ceqsal.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
43pm5.74i 271 . . 3 ((𝑥 = 𝐴𝜑) ↔ (𝑥 = 𝐴𝜓))
54albii 1818 . 2 (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∀𝑥(𝑥 = 𝐴𝜓))
6 ceqsal.2 . . . 4 𝐴 ∈ V
76isseti 3497 . . 3 𝑥 𝑥 = 𝐴
87a1bi 362 . 2 (𝜓 ↔ (∃𝑥 𝑥 = 𝐴𝜓))
92, 5, 83bitr4i 303 1 (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1537   = wceq 1539  wex 1778  wnf 1782  wcel 2107  Vcvv 3479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-12 2176
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783  df-clel 2815
This theorem is referenced by:  ceqsex  3529  ralxp3f  8163  aomclem6  43076
  Copyright terms: Public domain W3C validator