Theorem eqsb1 2336

Description: Substitution for the left-hand side in an equality. Class version of equsb3 2005. (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 2335	. . 3 |- ( [ w / x ] x = A <-> w = A )
2	1	sbbii 1814	. 2 |- ( [ y / w ] [ w / x ] x = A <-> [ y / w ] w = A )
3		nfv 1577	. . 3 |- F/ w x = A
4	3	sbco2 2019	. 2 |- ( [ y / w ] [ w / x ] x = A <-> [ y / x ] x = A )
5		eqsb1lem 2335	. 2 |- ( [ y / w ] w = A <-> y = A )
6	2, 4, 5	3bitr3i 210	1 |- ( [ y / x ] x = A <-> y = A )

Syntax hints:   <-> wb 105   = wceq 1398   [wsb 1811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-cleq 2225

This theorem is referenced by:  pm13.183  2955  eqsbc1  3082