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Theorem equsalvw 1977
 Description: Version of equsalv 2146 with a dv condition, and of equsal 2327 with two dv conditions, which requires fewer axioms. See also the dual form equsexvw 1978. (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 1911 . 2 (∀𝑥(𝑥 = 𝑦𝜓) ↔ (∃𝑥 𝑥 = 𝑦𝜓))
2 equsalvw.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32pm5.74i 260 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
43albii 1787 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜓))
5 ax6ev 1947 . . 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 1521  ∃wex 1744 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 This theorem depends on definitions:  df-bi 197  df-ex 1745 This theorem is referenced by:  ax13lem2  2332  reu8  3435  asymref2  5548  intirr  5549  fun11  6001  bj-dvelimdv  32959  bj-dvelimdv1  32960  wl-clelv2-just  33509  undmrnresiss  38227  pm13.192  38928
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