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Theorem equsalvw 2003
Description: Version of equsalv 2267 with a disjoint variable condition, and of equsal 2422 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsexvw 2004. (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 equsalvw.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
21pm5.74i 271 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
32albii 1819 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜓))
4 equsv 2002 . 2 (∀𝑥(𝑥 = 𝑦𝜓) ↔ 𝜓)
53, 4bitri 275 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967
This theorem depends on definitions:  df-bi 207  df-ex 1780
This theorem is referenced by:  equsexvw  2004  equvelv  2030  sb6  2085  sbievwOLD  2094  ax13lem2  2381  reu8  3739  el  5442  asymref2  6137  intirr  6138  fun11  6640  fv3  6924  fpwwe2lem11  10681  bj-dvelimdv  36852  bj-dvelimdv1  36853  undmrnresiss  43617  pm13.192  44429
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