Theorem equsalvw 2031

Description: Version of equsalv 2309 with a disjoint variable condition, and of equsal 2455 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsexvw 2032. (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 1846	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓))
4		equsv 2030	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜓)
5	3, 4	bitri 278	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994

This theorem depends on definitions:  df-bi 210  df-ex 1807

This theorem is referenced by:  equsexvw  2032  equvelv  2058  sb6  2125  sbievwOLD  2135  ax13lem2  2414  reu8  3705  elOLD  5423  asymref2  6120  intirr  6121  fun11  6613  fv3  6902  elirrvOLD  9562  fpwwe2lem11  10628  axprALT2  35448  axreg  35475  axregscl  35476  mh-prprimbi  36979  bj-dvelimdv  37411  bj-dvelimdv1  37412  undmrnresiss  44259  pm13.192  45049