Theorem eqsb1 2293

Description: Substitution for the left-hand side in an equality. Class version of equsb3 1963. (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 2292	. . 3 |- ( [ w / x ] x = A <-> w = A )
2	1	sbbii 1776	. 2 |- ( [ y / w ] [ w / x ] x = A <-> [ y / w ] w = A )
3		nfv 1539	. . 3 |- F/ w x = A
4	3	sbco2 1977	. 2 |- ( [ y / w ] [ w / x ] x = A <-> [ y / x ] x = A )
5		eqsb1lem 2292	. 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 1364   [wsb 1773

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-cleq 2182

This theorem is referenced by:  pm13.183  2890  eqsbc1  3017