Theorem equsalvw 2036

Description: Version of equsalv 2301 with a disjoint variable condition, and of equsal 2446 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsexvw 2037. (Contributed by BJ, 31-May-2019.)

Hypothesis

Ref	Expression
equsalvw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
equsalvw	⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑦)

Proof of Theorem equsalvw

Step	Hyp	Ref	Expression
1		equsalvw.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	pm5.74i 274	. . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜓))
3	2	albii 1851	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓))
4		equsv 2035	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜓)
5	3, 4	bitri 278	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  equsexvw  2037  equvelv  2063  sb6  2121  ax13lem2  2405  reu8  3690  el.OLD  5406  asymref2  6110  intirr  6111  fun11  6614  fv3  6903  elirrvOLD  9592  fpwwe2lem11  10726  axprALT2  35734  axreg  35795  axregscl  35796  mh-prprimbi  37331  bj-dvelimdv  37763  bj-dvelimdv1  37764  undmrnresiss  44604  pm13.192  45394