Theorem wl-ax11-lem1 34821

Description: A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.)

Assertion

Ref	Expression
wl-ax11-lem1	⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 ↔ ∀𝑦 𝑦 = 𝑧))

Proof of Theorem wl-ax11-lem1

Step	Hyp	Ref	Expression
1		wl-aetr 34773	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧))
2		wl-aetr 34773	. . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦 𝑦 = 𝑧 → ∀𝑥 𝑥 = 𝑧))
3	2	aecoms 2449	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦 𝑦 = 𝑧 → ∀𝑥 𝑥 = 𝑧))
4	1, 3	impbid 214	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 ↔ ∀𝑦 𝑦 = 𝑧))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-12 2176  ax-13 2389

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-nf 1784

This theorem is referenced by:  wl-ax11-lem8  34828