Theorem eq4 821

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
eq4	|- (A.x x = y -> A.y y = x)

Proof of Theorem eq4

Step	Hyp	Ref	Expression
1		ax-10 800	. . 3 |- (A.x x = y -> (A.x x = y -> A.y x = y))
2	1	pm2.43i 58	. 2 |- (A.x x = y -> A.y x = y)
3		eqcom 811	. . 3 |- (x = y -> y = x)
4	3	19.20i 691	. 2 |- (A.y x = y -> A.y y = x)
5	2, 4	syl 12	1 |- (A.x x = y -> A.y y = x)

Syntax hints:   -> wi 2  A.wal 672   = weq 797

This theorem is referenced by:  eq4s 822  eq4ds 823  axrepnd 3740

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-12 802

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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