Theorem eqsb3 2183

Description: Substitution applied to an atomic wff (class version of equsb3 1867). (Contributed by Rodolfo Medina, 28-Apr-2010.)

Assertion

Ref	Expression
eqsb3	|- ( [ x / y ] y = A <-> x = A )

Distinct variable group:   y, A

Allowed substitution hint:   A( x)

Proof of Theorem eqsb3

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqsb3lem 2182	. . 3 |- ( [ w / y ] y = A <-> w = A )
2	1	sbbii 1689	. 2 |- ( [ x / w ] [ w / y ] y = A <-> [ x / w ] w = A )
3		nfv 1462	. . 3 |- F/ w y = A
4	3	sbco2 1881	. 2 |- ( [ x / w ] [ w / y ] y = A <-> [ x / y ] y = A )
5		eqsb3lem 2182	. 2 |- ( [ x / w ] w = A <-> x = A )
6	2, 4, 5	3bitr3i 208	1 |- ( [ x / y ] y = A <-> x = A )

Syntax hints:   <-> wb 103   = wceq 1285   [wsb 1686

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-cleq 2075

This theorem is referenced by:  pm13.183  2733  eqsbc3  2854