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Theorem ceqsal 3504
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 2704. (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 2196 . 2 (∀𝑥(𝑥 = 𝐴𝜓) ↔ (∃𝑥 𝑥 = 𝐴𝜓))
3 ceqsal.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
43pm5.74i 271 . . 3 ((𝑥 = 𝐴𝜑) ↔ (𝑥 = 𝐴𝜓))
54albii 1813 . 2 (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∀𝑥(𝑥 = 𝐴𝜓))
6 ceqsal.2 . . . 4 𝐴 ∈ V
76isseti 3484 . . 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 205  wal 1531   = wceq 1533  wex 1773  wnf 1777  wcel 2098  Vcvv 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778  df-clel 2804
This theorem is referenced by:  ceqsalvOLD  3507  ceqsex  3518  ralxp3f  8120  aomclem6  42360
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