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Theorem ceqex 3584
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 2144 . . 3 ((𝑥 = 𝐴𝜑) → ∃𝑥(𝑥 = 𝐴𝜑))
21ex 413 . 2 (𝑥 = 𝐴 → (𝜑 → ∃𝑥(𝑥 = 𝐴𝜑)))
3 eqvisset 3454 . . . 4 (𝑥 = 𝐴𝐴 ∈ V)
4 alexeqg 3583 . . . 4 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
53, 4syl 17 . . 3 (𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
6 sp 2146 . . . 4 (∀𝑥(𝑥 = 𝐴𝜑) → (𝑥 = 𝐴𝜑))
76com12 32 . . 3 (𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴𝜑) → 𝜑))
85, 7sylbird 261 . 2 (𝑥 = 𝐴 → (∃𝑥(𝑥 = 𝐴𝜑) → 𝜑))
92, 8impbid 213 1 (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1520   = wceq 1522  wex 1761  wcel 2081  Vcvv 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-12 2141  ax-ext 2769
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-v 3439
This theorem is referenced by:  ceqsexg  3585
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