Theorem naecoms 2026

Description: A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)

Hypothesis

Ref	Expression
naecoms.1	|- ( -. A. x x = y -> ph )

Assertion

Ref	Expression
naecoms	|- ( -. A. y y = x -> ph )

Proof of Theorem naecoms

Step	Hyp	Ref	Expression
1		aecom 2024	. 2 |- ( A. x x = y <-> A. y y = x )
2		naecoms.1	. 2 |- ( -. A. x x = y -> ph )
3	1, 2	sylnbir 307	1 |- ( -. A. y y = x -> ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600

This theorem is referenced by:  sb9  2149  eujustALT  2278  nfcvf2  2655  axpowndlem2  8974  wl-sbcom2d  29864  wl-mo2dnae  29872