Theorem wl-equsalcom 34784

Description: This simple equivalence eases substitution of one expression for the other. (Contributed by Wolf Lammen, 1-Sep-2018.)

Assertion

Ref	Expression
wl-equsalcom	⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑦 = 𝑥 → 𝜑))

Proof of Theorem wl-equsalcom

Step	Hyp	Ref	Expression
1		equcom 2025	. . 3 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
2	1	imbi1i 352	. 2 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑦 = 𝑥 → 𝜑))
3	2	albii 1820	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑦 = 𝑥 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535

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 1911  ax-6 1970  ax-7 2015

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by:  wl-equsal1i  34785