Theorem equsb3 1925

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

Assertion

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

Distinct variable group:   x, z

Proof of Theorem equsb3

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equsb3lem 1924	. . 3 |- ( [ w / x ] x = z <-> w = z )
2	1	sbbii 1739	. 2 |- ( [ y / w ] [ w / x ] x = z <-> [ y / w ] w = z )
3		ax-17 1507	. . 3 |- ( x = z -> A. w x = z )
4	3	sbco2vh 1919	. 2 |- ( [ y / w ] [ w / x ] x = z <-> [ y / x ] x = z )
5		equsb3lem 1924	. 2 |- ( [ y / w ] w = z <-> y = z )
6	2, 4, 5	3bitr3i 209	1 |- ( [ y / x ] x = z <-> y = z )

Syntax hints:   <-> wb 104   [wsb 1736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737

This theorem is referenced by:  sb8eu  2013  sb8euh  2023  sb8iota  5103