Theorem eqsb1 2281

Description: Substitution for the left-hand side in an equality. Class version of equsb3 1951. (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 2280	. . 3 |- ( [ w / x ] x = A <-> w = A )
2	1	sbbii 1765	. 2 |- ( [ y / w ] [ w / x ] x = A <-> [ y / w ] w = A )
3		nfv 1528	. . 3 |- F/ w x = A
4	3	sbco2 1965	. 2 |- ( [ y / w ] [ w / x ] x = A <-> [ y / x ] x = A )
5		eqsb1lem 2280	. 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 1353   [wsb 1762

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-cleq 2170

This theorem is referenced by:  pm13.183  2876  eqsbc1  3003