Theorem nfsb2or 1837

Description: Bound-variable hypothesis builder for substitution. Similar to hbsb2 1836 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.)

Assertion

Ref	Expression
nfsb2or	⊢ (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Proof of Theorem nfsb2or

Step	Hyp	Ref	Expression
1		sb4or 1833	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		sb2 1767	. . . . . . 7 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)
3	2	a5i 1543	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥[𝑦 / 𝑥]𝜑)
4	3	imim2i 12	. . . . 5 ⊢ (([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
5	4	alimi 1455	. . . 4 ⊢ (∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
6		df-nf 1461	. . . 4 ⊢ (Ⅎ𝑥[𝑦 / 𝑥]𝜑 ↔ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
7	5, 6	sylibr 134	. . 3 ⊢ (∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
8	7	orim2i 761	. 2 ⊢ ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) → (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥[𝑦 / 𝑥]𝜑))
9	1, 8	ax-mp 5	1 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ∨ wo 708  ∀wal 1351  Ⅎwnf 1460  [wsb 1762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by:  sbequi  1839