Theorem bj-sbime 13808

Description: A strengthening of sbie 1784 (same proof). (Contributed by BJ, 16-Dec-2019.)

Hypotheses

Ref	Expression
bj-sbime.nf	⊢ Ⅎ𝑥𝜓
bj-sbime.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
bj-sbime	⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)

Proof of Theorem bj-sbime

Step	Hyp	Ref	Expression
1		bj-sbime.nf	. . 3 ⊢ Ⅎ𝑥𝜓
2	1	nfri 1512	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
3		bj-sbime.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	bj-sbimeh 13807	1 ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)

Syntax hints:   → wi 4  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-ial 1527

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756

This theorem is referenced by:  setindis  14002  bdsetindis  14004