Theorem aecoms 2429

Description: A commutation rule for identical variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 2373. (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 2428	. 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 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-10 2140  ax-12 2174  ax-13 2373

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786  df-nf 1790

This theorem is referenced by:  axc11  2431  nd4  10330  axrepnd  10334  axpownd  10341  axregnd  10344  axinfnd  10346  axacndlem5  10351  axacnd  10352  wl-ax11-lem1  35715  wl-ax11-lem3  35717  wl-ax11-lem9  35723  wl-ax11-lem10  35724  e2ebind  42136