Theorem bj-sbimeh 10583

Description: A strengthening of sbieh 1713 (same proof). (Contributed by BJ, 16-Dec-2019.)

Hypotheses

Ref	Expression
bj-sbimeh.1	⊢ (𝜓 → ∀𝑥𝜓)
bj-sbimeh.2	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

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

Proof of Theorem bj-sbimeh

Step	Hyp	Ref	Expression
1		tru 1288	. . . 4 ⊢ ⊤
2	1	hbth 1392	. . 3 ⊢ (⊤ → ∀𝑥⊤)
3		bj-sbimeh.1	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
4	3	a1i 9	. . 3 ⊢ (⊤ → (𝜓 → ∀𝑥𝜓))
5		bj-sbimeh.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
6	5	a1i 9	. . 3 ⊢ (⊤ → (𝑥 = 𝑦 → (𝜑 → 𝜓)))
7	2, 4, 6	bj-sbimedh 10582	. 2 ⊢ (⊤ → ([𝑦 / 𝑥]𝜑 → 𝜓))
8	7	trud 1293	1 ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1282  ⊤wtru 1285  [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-sb 1686

This theorem is referenced by:  bj-sbime  10584