Theorem aecoms 2428

Description: A commutation rule for identical variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 2372. (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 2427	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
2		aecoms.1	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)
3	1, 2	sylbi 216	1 ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-nf 1787

This theorem is referenced by:  axc11  2430  nd4  10346  axrepnd  10350  axpownd  10357  axregnd  10360  axinfnd  10362  axacndlem5  10367  axacnd  10368  wl-ax11-lem1  35736  wl-ax11-lem3  35738  wl-ax11-lem9  35744  wl-ax11-lem10  35745  e2ebind  42183