Theorem eqsb1 2860

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		eqeq1 2737	. 2 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝐴 ↔ 𝑤 = 𝐴))
2		eqeq1 2737	. 2 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝐴 ↔ 𝑦 = 𝐴))
3	1, 2	sbievw2 2100	1 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴)

Syntax hints:   ↔ wb 205   = wceq 1542  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-sb 2069  df-cleq 2725

This theorem is referenced by:  sbhypf  3539  pm13.183  3657  eqsbc1  3827