Theorem equsb3 1963

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 1962	. . 3 |- ( [ w / x ] x = z <-> w = z )
2	1	sbbii 1776	. 2 |- ( [ y / w ] [ w / x ] x = z <-> [ y / w ] w = z )
3		ax-17 1537	. . 3 |- ( x = z -> A. w x = z )
4	3	sbco2vh 1957	. 2 |- ( [ y / w ] [ w / x ] x = z <-> [ y / x ] x = z )
5		equsb3lem 1962	. 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 1773

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774

This theorem is referenced by:  sb8eu  2051  sb8euh  2061  sb8iota  5210