Theorem eqsb3 2728

Description: Substitution applied to an atomic wff (class version of equsb3 2432). (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 2727	. . 3 |- ( [ w / y ] y = A <-> w = A )
2	1	sbbii 1887	. 2 |- ( [ x / w ] [ w / y ] y = A <-> [ x / w ] w = A )
3		nfv 1843	. . 3 |- F/ w y = A
4	3	sbco2 2415	. 2 |- ( [ x / w ] [ w / y ] y = A <-> [ x / y ] y = A )
5		eqsb3lem 2727	. 2 |- ( [ x / w ] w = A <-> x = A )
6	2, 4, 5	3bitr3i 290	1 |- ( [ x / y ] y = A <-> x = A )

Syntax hints:   <-> wb 196   = wceq 1483   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-cleq 2615

This theorem is referenced by:  pm13.183  3344  eqsbc3  3475