Theorem equsb3 1328

Description: Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)

Assertion

Ref	Expression
equsb3	|- ([x / y]y = z <-> x = z)

Distinct variable group:   y,z

Proof of Theorem equsb3

Step	Hyp	Ref	Expression
1		equsb3lem 1327	. . 3 |- ([w / y]y = z <-> w = z)
2	1	sbbii 1172	. 2 |- ([x / w][w / y]y = z <-> [x / w]w = z)
3		ax-17 969	. . 3 |- (y = z -> A.w y = z)
4	3	sbco2 1253	. 2 |- ([x / w][w / y]y = z <-> [x / y]y = z)
5		equsb3lem 1327	. 2 |- ([x / w]w = z <-> x = z)
6	2, 4, 5	3bitr3 181	1 |- ([x / y]y = z <-> x = z)

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

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-9 963  ax-10 964  ax-11 965  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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