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Theorem equsalvw 2103
 Description: Version of equsalv 2290 with a disjoint variable condition, and of equsal 2424 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsexvw 2104. (Contributed by BJ, 31-May-2019.)
Hypothesis
Ref Expression
equsalvw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
equsalvw (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Distinct variable groups:   𝑥,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem equsalvw
StepHypRef Expression
1 19.23v 2038 . 2 (∀𝑥(𝑥 = 𝑦𝜓) ↔ (∃𝑥 𝑥 = 𝑦𝜓))
2 equsalvw.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32pm5.74i 263 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
43albii 1915 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜓))
5 ax6ev 2074 . . 3 𝑥 𝑥 = 𝑦
65a1bi 354 . 2 (𝜓 ↔ (∃𝑥 𝑥 = 𝑦𝜓))
71, 4, 63bitr4i 295 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1651  ∃wex 1875 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072 This theorem depends on definitions:  df-bi 199  df-ex 1876 This theorem is referenced by:  equvelv  2132  ax13lem2  2381  reu8  3598  asymref2  5731  intirr  5732  fun11  6174  fpwwe2lem12  9751  bj-dvelimdv  33328  bj-dvelimdv1  33329  wl-clelv2-just  33868  undmrnresiss  38693  pm13.192  39392
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