Theorem equsb3 1939

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 1938	. . 3 |- ( [ w / x ] x = z <-> w = z )
2	1	sbbii 1753	. 2 |- ( [ y / w ] [ w / x ] x = z <-> [ y / w ] w = z )
3		ax-17 1514	. . 3 |- ( x = z -> A. w x = z )
4	3	sbco2vh 1933	. 2 |- ( [ y / w ] [ w / x ] x = z <-> [ y / x ] x = z )
5		equsb3lem 1938	. 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 1750

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by:  sb8eu  2027  sb8euh  2037  sb8iota  5160