Theorem wl-ax11-lem9 35772

Description: The easy part when 𝑥 coincides with 𝑦. (Contributed by Wolf Lammen, 30-Jun-2019.)

Assertion

Ref	Expression
wl-ax11-lem9	⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑))

Proof of Theorem wl-ax11-lem9

Step	Hyp	Ref	Expression
1		biidd 261	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
2	1	dral1 2434	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜑))
3	2	aecoms 2423	. . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 ↔ ∀𝑦𝜑))
4	3	dral1 2434	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑))
5	4	aecoms 2423	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦∀𝑥𝜑 ↔ ∀𝑥∀𝑦𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2132  ax-12 2166  ax-13 2367

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1778  df-nf 1782

This theorem is referenced by: (None)