Theorem sb4or 1806

Description: One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1805 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.)

Assertion

Ref	Expression
sb4or	⊢ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Proof of Theorem sb4or

Step	Hyp	Ref	Expression
1		equs5or 1803	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		nfe1 1473	. . . . . 6 ⊢ Ⅎ𝑥∃𝑥(𝑥 = 𝑦 ∧ 𝜑)
3		nfa1 1522	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → 𝜑)
4	2, 3	nfim 1552	. . . . 5 ⊢ Ⅎ𝑥(∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
5	4	nfri 1500	. . . 4 ⊢ ((∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∀𝑥(∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
6		sb1 1740	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
7	6	imim1i 60	. . . 4 ⊢ ((∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
8	5, 7	alrimih 1446	. . 3 ⊢ ((∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
9	8	orim2i 751	. 2 ⊢ ((∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))) → (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
10	1, 9	ax-mp 5	1 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 698  ∀wal 1330  ∃wex 1469  [wsb 1736

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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737

This theorem is referenced by:  sb4bor  1808  nfsb2or  1810