Theorem bj-sbime 10584

Description: A strengthening of sbie 1714 (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 1452	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
3		bj-sbime.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	bj-sbimeh 10583	1 ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1389  [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686

This theorem is referenced by:  setindis  10762  bdsetindis  10764