Theorem bj-sbime 10735

Description: A strengthening of sbie 1715 (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 1453	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
3		bj-sbime.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	bj-sbimeh 10734	1 ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1390  [wsb 1686

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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687

This theorem is referenced by:  setindis  10920  bdsetindis  10922