Theorem aecoms 2394

Description: A commutation rule for identical variable specifiers. (Contributed by NM, 10-May-1993.)

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 2393	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
2		aecoms.1	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑)
3	1, 2	sylbi 209	1 ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-12 2163  ax-13 2334

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-nf 1828

This theorem is referenced by:  axc11  2396  nd4  9749  axrepnd  9753  axpownd  9760  axregnd  9763  axinfnd  9765  axacndlem5  9770  axacnd  9771  wl-ax11-lem1  33959  wl-ax11-lem3  33961  wl-ax11-lem9  33967  wl-ax11-lem10  33968  e2ebind  39733