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Theorem equsalvw 2089
Description: Version of equsalv 2264 with a dv condition, and of equsal 2446 with two dv conditions, which requires fewer axioms. See also the dual form equsexvw 2090. (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 2023 . 2 (∀𝑥(𝑥 = 𝑦𝜓) ↔ (∃𝑥 𝑥 = 𝑦𝜓))
2 equsalvw.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32pm5.74i 260 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
43albii 1895 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜓))
5 ax6ev 2059 . . 3 𝑥 𝑥 = 𝑦
65a1bi 351 . 2 (𝜓 ↔ (∃𝑥 𝑥 = 𝑦𝜓))
71, 4, 63bitr4i 292 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057
This theorem depends on definitions:  df-bi 197  df-ex 1853
This theorem is referenced by:  equvelv  2118  ax13lem2  2451  reu8  3554  asymref2  5652  intirr  5653  fun11  6101  bj-dvelimdv  33161  bj-dvelimdv1  33162  wl-clelv2-just  33705  undmrnresiss  38429  pm13.192  39130
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