Theorem eqsb1 2311

Description: Substitution for the left-hand side in an equality. Class version of equsb3 1980. (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 2310	. . 3 |- ( [ w / x ] x = A <-> w = A )
2	1	sbbii 1789	. 2 |- ( [ y / w ] [ w / x ] x = A <-> [ y / w ] w = A )
3		nfv 1552	. . 3 |- F/ w x = A
4	3	sbco2 1994	. 2 |- ( [ y / w ] [ w / x ] x = A <-> [ y / x ] x = A )
5		eqsb1lem 2310	. 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 1373   [wsb 1786

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-cleq 2200

This theorem is referenced by:  pm13.183  2918  eqsbc1  3045