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Theorem ceqex 3327
Description: Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.) (Proof shortened by BJ, 1-May-2019.)
Assertion
Ref Expression
ceqex (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqex
StepHypRef Expression
1 19.8a 2050 . . 3 ((𝑥 = 𝐴𝜑) → ∃𝑥(𝑥 = 𝐴𝜑))
21ex 450 . 2 (𝑥 = 𝐴 → (𝜑 → ∃𝑥(𝑥 = 𝐴𝜑)))
3 eqvisset 3206 . . . 4 (𝑥 = 𝐴𝐴 ∈ V)
4 alexeqg 3326 . . . 4 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
53, 4syl 17 . . 3 (𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
6 sp 2051 . . . 4 (∀𝑥(𝑥 = 𝐴𝜑) → (𝑥 = 𝐴𝜑))
76com12 32 . . 3 (𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴𝜑) → 𝜑))
85, 7sylbird 250 . 2 (𝑥 = 𝐴 → (∃𝑥(𝑥 = 𝐴𝜑) → 𝜑))
92, 8impbid 202 1 (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1479   = wceq 1481  wex 1702  wcel 1988  Vcvv 3195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-12 2045  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-v 3197
This theorem is referenced by:  ceqsexg  3328
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