Theorem eqsb3 2549

Description: Substitution applied to an atomic wff (class version of equsb3 2228). (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 2548	. . 3 |- ( [ w / y ] y = A <-> w = A )
2	1	sbbii 1796	. 2 |- ( [ x / w ] [ w / y ] y = A <-> [ x / w ] w = A )
3		nfv 1754	. . 3 |- F/ w y = A
4	3	sbco2 2210	. 2 |- ( [ x / w ] [ w / y ] y = A <-> [ x / y ] y = A )
5		eqsb3lem 2548	. 2 |- ( [ x / w ] w = A <-> x = A )
6	2, 4, 5	3bitr3i 278	1 |- ( [ x / y ] y = A <-> x = A )

Syntax hints:   <-> wb 187   = wceq 1437   [wsb 1789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-ex 1660  df-nf 1664  df-sb 1790  df-cleq 2421

This theorem is referenced by:  pm13.183  3218  eqsbc3  3345