Theorem aecom-o 37766

Description: Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). Version of aecom 2426 using ax-c11 37752. Unlike axc11nfromc11 37791, this version does not require ax-5 1913 (see comment of equcomi1 37765). (Contributed by NM, 10-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
aecom-o	⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Proof of Theorem aecom-o

Step	Hyp	Ref	Expression
1		ax-c11 37752	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
2	1	pm2.43i 52	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3		equcomi1 37765	. . 3 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
4	3	alimi 1813	. 2 ⊢ (∀𝑦 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
5	2, 4	syl 17	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Syntax hints:   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-c5 37748  ax-c4 37749  ax-c7 37750  ax-c10 37751  ax-c11 37752  ax-c9 37755

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782

This theorem is referenced by:  aecoms-o  37767  naecoms-o  37792  aev-o  37796  ax12indalem  37810