Theorem aecoms 2438

Description: A commutation rule for identical variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 2382. (Contributed by NM, 10-May-1993.) (New usage is discouraged.)

Hypothesis

Ref	Expression
aecoms.1	⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)

Assertion

Ref	Expression
aecoms	⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)

Proof of Theorem aecoms

Step	Hyp	Ref	Expression
1		aecom 2437	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
2		aecoms.1	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)
3	1, 2	sylbi 219	1 ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-10 2154  ax-12 2191  ax-13 2382

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-nf 1792

This theorem is referenced by:  axc11  2440  nd4  10508  axrepnd  10512  axpownd  10519  axregnd  10522  axinfnd  10524  axacndlem5  10529  axacnd  10530  e2ebind  45022