Theorem eqsb1 2888

Description: Substitution for the left-hand side in an equality. Class version of equsb3 2137. (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 2766	. 2 ⊢ (𝑥 = 𝑤 → (𝑥 = 𝐴 ↔ 𝑤 = 𝐴))
2		eqeq1 2766	. 2 ⊢ (𝑤 = 𝑦 → (𝑤 = 𝐴 ↔ 𝑦 = 𝐴))
3	1, 2	sbievw2 2132	1 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  [wsb 2095

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-sb 2096  df-cleq 2754

This theorem is used by:  sbhypf  3513  pm13.183  3624  eqsbc1  3789