Theorem equsb3 1880

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 1879	. . 3 |- ( [ w / y ] y = z <-> w = z )
2	1	sbbii 1702	. 2 |- ( [ x / w ] [ w / y ] y = z <-> [ x / w ] w = z )
3		ax-17 1471	. . 3 |- ( y = z -> A. w y = z )
4	3	sbco2v 1876	. 2 |- ( [ x / w ] [ w / y ] y = z <-> [ x / y ] y = z )
5		equsb3lem 1879	. 2 |- ( [ x / w ] w = z <-> x = z )
6	2, 4, 5	3bitr3i 209	1 |- ( [ x / y ] y = z <-> x = z )

Syntax hints:   <-> wb 104   [wsb 1699

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700

This theorem is referenced by:  sb8eu  1968  sb8euh  1978  sb8iota  5021