Theorem nfsb2or 1715

Description: Bound-variable hypothesis builder for substitution. Similar to hbsb2 1714 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 1711	. 2 ⊢ (∀x x = y ∨ ∀x([y / x]φ → ∀x(x = y → φ)))
2		sb2 1647	. . . . . . 7 ⊢ (∀x(x = y → φ) → [y / x]φ)
3	2	a5i 1432	. . . . . 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 1341	. . . 4 ⊢ (∀x([y / x]φ → ∀x(x = y → φ)) → ∀x([y / x]φ → ∀x[y / x]φ))
6		df-nf 1347	. . . 4 ⊢ (Ⅎx[y / x]φ ↔ ∀x([y / x]φ → ∀x[y / x]φ))
7	5, 6	sylibr 137	. . 3 ⊢ (∀x([y / x]φ → ∀x(x = y → φ)) → Ⅎx[y / x]φ)
8	7	orim2i 677	. 2 ⊢ ((∀x x = y ∨ ∀x([y / x]φ → ∀x(x = y → φ))) → (∀x x = y ∨ Ⅎx[y / x]φ))
9	1, 8	ax-mp 7	1 ⊢ (∀x x = y ∨ Ⅎx[y / x]φ)

Syntax hints:   → wi 4   ∨ wo 628  ∀wal 1240  Ⅎwnf 1346  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by:  sbequi  1717