Theorem equsb3lem 1327

Description: Lemma for equsb3 1328.

Assertion

Ref	Expression
equsb3lem	|- ([x / y]y = z <-> x = z)

Distinct variable groups:   y,z   x,y

Proof of Theorem equsb3lem

Step	Hyp	Ref	Expression
1		equsb2 1192	. . . 4 |- [x / y]x = y
2		equequ1 1132	. . . . 5 |- (x = y -> (x = z <-> y = z))
3	2	sbimi 1171	. . . 4 |- ([x / y]x = y -> [x / y](x = z <-> y = z))
4	1, 3	ax-mp 7	. . 3 |- [x / y](x = z <-> y = z)
5		sbbi 1237	. . 3 |- ([x / y](x = z <-> y = z) <-> ([x / y]x = z <-> [x / y]y = z))
6	4, 5	mpbi 189	. 2 |- ([x / y]x = z <-> [x / y]y = z)
7		ax-17 969	. . 3 |- (x = z -> A.y x = z)
8	7	sbf 1184	. 2 |- ([x / y]x = z <-> x = z)
9	6, 8	bitr3 175	1 |- ([x / y]y = z <-> x = z)

Syntax hints:   <-> wb 146   = wceq 954  [wsbc 1168

This theorem is referenced by:  equsb3 1328

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170

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