Theorem eqsb1 2454

Description: Substitution for the left-hand side in an equality. Class version of equsb3 2102. (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 2453	. . 3 ⊢ ([w / x]x = A ↔ w = A)
2	1	sbbii 1653	. 2 ⊢ ([y / w][w / x]x = A ↔ [y / w]w = A)
3		nfv 1619	. . 3 ⊢ Ⅎw x = A
4	3	sbco2 2086	. 2 ⊢ ([y / w][w / x]x = A ↔ [y / x]x = A)
5		eqsb1lem 2453	. 2 ⊢ ([y / w]w = A ↔ y = A)
6	2, 4, 5	3bitr3i 266	1 ⊢ ([y / x]x = A ↔ y = A)

Syntax hints:   ↔ wb 176   = wceq 1642  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346

This theorem is referenced by:  pm13.183  2980  eqsbc1  3086