Theorem equsalvw 2089

Description: Version of equsalv 2264 with a dv condition, and of equsal 2446 with two dv conditions, which requires fewer axioms. See also the dual form equsexvw 2090. (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 2023	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓))
2		equsalvw.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	pm5.74i 260	. . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜓))
4	3	albii 1895	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓))
5		ax6ev 2059	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
6	5	a1bi 351	. 2 ⊢ (𝜓 ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓))
7	1, 4, 6	3bitr4i 292	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1629  ∃wex 1852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057

This theorem depends on definitions:  df-bi 197  df-ex 1853

This theorem is referenced by:  equvelv  2118  ax13lem2  2451  reu8  3554  asymref2  5652  intirr  5653  fun11  6101  bj-dvelimdv  33161  bj-dvelimdv1  33162  wl-clelv2-just  33705  undmrnresiss  38429  pm13.192  39130