Theorem equsb3 1330

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 1329	. . 3 |- ([w / y]y = z <-> w = z)
2	1	sbbii 1174	. 2 |- ([x / w][w / y]y = z <-> [x / w]w = z)
3		ax-17 970	. . 3 |- (y = z -> A.w y = z)
4	3	sbco2 1255	. 2 |- ([x / w][w / y]y = z <-> [x / y]y = z)
5		equsb3lem 1329	. 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 955  [wsbc 1170

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172

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