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Theorem ceqsexg 3317
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 2024 . . 3 𝑥𝑥(𝑥 = 𝐴𝜑)
2 ceqsexg.1 . . 3 𝑥𝜓
31, 2nfbi 1830 . 2 𝑥(∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
4 ceqex 3316 . . 3 (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
5 ceqsexg.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
64, 5bibi12d 335 . 2 (𝑥 = 𝐴 → ((𝜑𝜑) ↔ (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)))
7 biid 251 . 2 (𝜑𝜑)
83, 6, 7vtoclg1f 3251 1 (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wnf 1705  wcel 1987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-12 2044  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3188
This theorem is referenced by:  ceqsexgv  3318
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