Theorem sbiev 1785

Description: Conversion of implicit substitution to explicit substitution. Version of sbie 1784 with a disjoint variable condition. (Contributed by Wolf Lammen, 18-Jan-2023.)

Hypotheses

Ref	Expression
sbiev.1	⊢ Ⅎ𝑥𝜓
sbiev.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbiev	⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sbiev

Step	Hyp	Ref	Expression
1		sbiev.1	. 2 ⊢ Ⅎ𝑥𝜓
2		sbiev.2	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	sbie 1784	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 104  Ⅎwnf 1453  [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  sbco2v  1941  cbvabw  2293  csbcow  3060