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Theorem equsalvw 2006
Description: Version of equsalv 2257 with a disjoint variable condition, and of equsal 2415 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsexvw 2007. (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 270 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
32albii 1820 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜓))
4 equsv 2005 . 2 (∀𝑥(𝑥 = 𝑦𝜓) ↔ 𝜓)
53, 4bitri 274 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970
This theorem depends on definitions:  df-bi 206  df-ex 1781
This theorem is referenced by:  equsexvw  2007  equvelv  2033  sb6  2087  sbievw  2094  ax13lem2  2374  reu8  3730  el  5438  asymref2  6119  intirr  6120  fun11  6623  fv3  6910  fpwwe2lem11  10639  bj-dvelimdv  36034  bj-dvelimdv1  36035  undmrnresiss  42658  pm13.192  43472
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