Theorem equsb3 1867

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

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equsb3lem 1866	. . 3 |- ( [ w / y ] y = z <-> w = z )
2	1	sbbii 1689	. 2 |- ( [ x / w ] [ w / y ] y = z <-> [ x / w ] w = z )
3		ax-17 1460	. . 3 |- ( y = z -> A. w y = z )
4	3	sbco2v 1863	. 2 |- ( [ x / w ] [ w / y ] y = z <-> [ x / y ] y = z )
5		equsb3lem 1866	. 2 |- ( [ x / w ] w = z <-> x = z )
6	2, 4, 5	3bitr3i 208	1 |- ( [ x / y ] y = z <-> x = z )

Syntax hints:   <-> wb 103   [wsb 1686

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687

This theorem is referenced by:  sb8eu  1955  sb8euh  1965  sb8iota  4904