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Theorem ceqsexg 3556
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)
Hypotheses
Ref Expression
ceqsexg.1 𝑥𝜓
ceqsexg.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsexg (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem ceqsexg
StepHypRef Expression
1 nfe1 2088 . . 3 𝑥𝑥(𝑥 = 𝐴𝜑)
2 ceqsexg.1 . . 3 𝑥𝜓
31, 2nfbi 1867 . 2 𝑥(∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
4 ceqex 3555 . . 3 (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
5 ceqsexg.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
64, 5bibi12d 338 . 2 (𝑥 = 𝐴 → ((𝜑𝜑) ↔ (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)))
7 biid 253 . 2 (𝜑𝜑)
83, 6, 7vtoclg1f 3480 1 (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1508  wex 1743  wnf 1747  wcel 2051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-12 2107  ax-ext 2745
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-v 3412
This theorem is referenced by:  ceqsexgv  3557
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