Theorem eqsb3 2556

Description: Substitution applied to an atomic wff (class version of equsb3 2261). (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 2555	. . 3 |- ( [ w / y ] y = A <-> w = A )
2	1	sbbii 1804	. 2 |- ( [ x / w ] [ w / y ] y = A <-> [ x / w ] w = A )
3		nfv 1761	. . 3 |- F/ w y = A
4	3	sbco2 2244	. 2 |- ( [ x / w ] [ w / y ] y = A <-> [ x / y ] y = A )
5		eqsb3lem 2555	. 2 |- ( [ x / w ] w = A <-> x = A )
6	2, 4, 5	3bitr3i 279	1 |- ( [ x / y ] y = A <-> x = A )

Syntax hints:   <-> wb 188   = wceq 1444   [wsb 1797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-ex 1664  df-nf 1668  df-sb 1798  df-cleq 2444

This theorem is referenced by:  pm13.183  3179  eqsbc3  3307