Theorem equsb3 1951

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 1950	. . 3 |- ( [ w / x ] x = z <-> w = z )
2	1	sbbii 1765	. 2 |- ( [ y / w ] [ w / x ] x = z <-> [ y / w ] w = z )
3		ax-17 1526	. . 3 |- ( x = z -> A. w x = z )
4	3	sbco2vh 1945	. 2 |- ( [ y / w ] [ w / x ] x = z <-> [ y / x ] x = z )
5		equsb3lem 1950	. 2 |- ( [ y / w ] w = z <-> y = z )
6	2, 4, 5	3bitr3i 210	1 |- ( [ y / x ] x = z <-> y = z )

Syntax hints:   <-> wb 105   [wsb 1762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by:  sb8eu  2039  sb8euh  2049  sb8iota  5183