Theorem alequcom 1125

Description: Commutation law for identical variable specifiers. The antecedent and consequent are true when x and y are substituted with the same variable.

Assertion

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

Proof of Theorem alequcom

Step	Hyp	Ref	Expression
1		ax-10 1103	. . 3 |- (A.x x = y -> (A.x x = y -> A.y x = y))
2	1	pm2.43i 64	. 2 |- (A.x x = y -> A.y x = y)
3		equcomi 1115	. . 3 |- (x = y -> y = x)
4	3	19.20i 968	. 2 |- (A.y x = y -> A.y y = x)
5	2, 4	syl 10	1 |- (A.x x = y -> A.y y = x)

Syntax hints:   -> wi 3  A.wal 950   = wceq 1099

This theorem is referenced by:  alequcoms 1126  nalequcoms 1127  aev 1192  ax11indalem 1345  a12stdy2 1350  axrepnd 4869

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957

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