Theorem aecom-o 34186

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 2311 using ax-c11 34172. Unlike axc11nfromc11 34211, this version does not require ax-5 1839 (see comment of equcomi1 34185). (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 34172	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦))
2	1	pm2.43i 52	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑥 = 𝑦)
3		equcomi1 34185	. . 3 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
4	3	alimi 1739	. 2 ⊢ (∀𝑦 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
5	2, 4	syl 17	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Syntax hints:   → wi 4  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-c5 34168  ax-c4 34169  ax-c7 34170  ax-c10 34171  ax-c11 34172  ax-c9 34175

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  aecoms-o  34187  naecoms-o  34212  aev-o  34216  ax12indalem  34230