Theorem eqsb1 2274

Description: Substitution for the left-hand side in an equality. Class version of equsb3 1944. (Contributed by Rodolfo Medina, 28-Apr-2010.)

Assertion

Ref	Expression
eqsb1	|- ( [ y / x ] x = A <-> y = A )

Distinct variable group:   x, A

Allowed substitution hint:   A( y)

Proof of Theorem eqsb1

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqsb1lem 2273	. . 3 |- ( [ w / x ] x = A <-> w = A )
2	1	sbbii 1758	. 2 |- ( [ y / w ] [ w / x ] x = A <-> [ y / w ] w = A )
3		nfv 1521	. . 3 |- F/ w x = A
4	3	sbco2 1958	. 2 |- ( [ y / w ] [ w / x ] x = A <-> [ y / x ] x = A )
5		eqsb1lem 2273	. 2 |- ( [ y / w ] w = A <-> y = A )
6	2, 4, 5	3bitr3i 209	1 |- ( [ y / x ] x = A <-> y = A )

Syntax hints:   <-> wb 104   = wceq 1348   [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-cleq 2163

This theorem is referenced by:  pm13.183  2868  eqsbc1  2994