Theorem bj-sbimeh 13653

Description: A strengthening of sbieh 1778 (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 1347	. . . 4 ⊢ ⊤
2	1	hbth 1451	. . 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 13652	. 2 ⊢ (⊤ → ([𝑦 / 𝑥]𝜑 → 𝜓))
8	7	mptru 1352	1 ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1341  ⊤wtru 1344  [wsb 1750

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-sb 1751

This theorem is referenced by:  bj-sbime  13654