Theorem naecoms-o 34212

Description: A commutation rule for distinct variable specifiers. Version of naecoms 2313 using ax-c11 34172. (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 34186	. . 3 |- ( A. x x = y -> A. y y = x )
2		nalequcoms-o.1	. . 3 |- ( -. A. x x = y -> ph )
3	1, 2	nsyl4 156	. 2 |- ( -. ph -> A. y y = x )
4	3	con1i 144	1 |- ( -. A. y y = x -> ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-c5 34168  ax-c4 34169  ax-c7 34170  ax-c10 34171  ax-c11 34172  ax-c9 34175

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  ax12inda2ALT  34231