Theorem naecoms-o 2178

Description: A commutation rule for distinct variable specifiers. Version of naecoms 1948 using ax-10o 2139. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

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

Proof of Theorem naecoms-o

Step	Hyp	Ref	Expression
1		aecom-o 2151	. . 3 |- (A.x x = y -> A.y y = x)
2		nalequcoms-o.1	. . 3 |- (-. A.x x = y -> ph)
3	1, 2	nsyl4 134	. 2 |- (-. ph -> A.y y = x)
4	3	con1i 121	1 |- (-. A.y y = x -> ph)

Syntax hints:  -. wn 3   -> wi 4  A.wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-10o 2139

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  ax11inda2ALT  2198