Theorem aecoms 2426

Description: A commutation rule for identical variable specifiers. Usage of this theorem is discouraged because it depends on ax-13 2370. (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 2425	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
2		aecoms.1	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)
3	1, 2	sylbi 217	1 ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-10 2142  ax-12 2178  ax-13 2370

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784

This theorem is referenced by:  axc11  2428  nd4  10543  axrepnd  10547  axpownd  10554  axregnd  10557  axinfnd  10559  axacndlem5  10564  axacnd  10565  wl-ax11-lem1  37573  wl-ax11-lem3  37575  wl-ax11-lem9  37581  wl-ax11-lem10  37582  e2ebind  44553