Theorem equsalvw 2103

Description: Version of equsalv 2290 with a disjoint variable condition, and of equsal 2424 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsexvw 2104. (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		19.23v 2038	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓))
2		equsalvw.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	pm5.74i 263	. . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜓))
4	3	albii 1915	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓))
5		ax6ev 2074	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
6	5	a1bi 354	. 2 ⊢ (𝜓 ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓))
7	1, 4, 6	3bitr4i 295	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1651  ∃wex 1875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072

This theorem depends on definitions:  df-bi 199  df-ex 1876

This theorem is referenced by:  equvelv  2132  ax13lem2  2381  reu8  3598  asymref2  5731  intirr  5732  fun11  6174  fpwwe2lem12  9751  bj-dvelimdv  33328  bj-dvelimdv1  33329  wl-clelv2-just  33868  undmrnresiss  38693  pm13.192  39392