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Theorem ceqex 3364
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 2090 . . 3 ((𝑥 = 𝐴𝜑) → ∃𝑥(𝑥 = 𝐴𝜑))
21ex 449 . 2 (𝑥 = 𝐴 → (𝜑 → ∃𝑥(𝑥 = 𝐴𝜑)))
3 eqvisset 3242 . . . 4 (𝑥 = 𝐴𝐴 ∈ V)
4 alexeqg 3363 . . . 4 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
53, 4syl 17 . . 3 (𝑥 = 𝐴 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
6 sp 2091 . . . 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 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-12 2087  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233
This theorem is referenced by:  ceqsexg  3365
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