Theorem axi12 2333

Description: Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12).

In classical logic, this is mostly a restatement of ax12o 1934 (with one additional quantifier). But in intuitionistic logic, changing the negations and implications to disjunctions makes it stronger.

(Contributed by Jim Kingdon, 31-Dec-2017.)

Assertion

Ref	Expression
axi12	⊢ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y)))

Proof of Theorem axi12

Step	Hyp	Ref	Expression
1		ax12o 1934	. . . . . . 7 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y)))
2		df-or 359	. . . . . . . 8 ⊢ ((∀z z = y ∨ (x = y → ∀z x = y)) ↔ (¬ ∀z z = y → (x = y → ∀z x = y)))
3	2	imbi2i 303	. . . . . . 7 ⊢ ((¬ ∀z z = x → (∀z z = y ∨ (x = y → ∀z x = y))) ↔ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y))))
4	1, 3	mpbir 200	. . . . . 6 ⊢ (¬ ∀z z = x → (∀z z = y ∨ (x = y → ∀z x = y)))
5		df-or 359	. . . . . 6 ⊢ ((∀z z = x ∨ (∀z z = y ∨ (x = y → ∀z x = y))) ↔ (¬ ∀z z = x → (∀z z = y ∨ (x = y → ∀z x = y))))
6	4, 5	mpbir 200	. . . . 5 ⊢ (∀z z = x ∨ (∀z z = y ∨ (x = y → ∀z x = y)))
7		orass 510	. . . . 5 ⊢ (((∀z z = x ∨ ∀z z = y) ∨ (x = y → ∀z x = y)) ↔ (∀z z = x ∨ (∀z z = y ∨ (x = y → ∀z x = y))))
8	6, 7	mpbir 200	. . . 4 ⊢ ((∀z z = x ∨ ∀z z = y) ∨ (x = y → ∀z x = y))
9	8	ax-gen 1546	. . 3 ⊢ ∀z((∀z z = x ∨ ∀z z = y) ∨ (x = y → ∀z x = y))
10		nfa1 1788	. . . . 5 ⊢ Ⅎz∀z z = x
11		nfa1 1788	. . . . 5 ⊢ Ⅎz∀z z = y
12	10, 11	nfor 1836	. . . 4 ⊢ Ⅎz(∀z z = x ∨ ∀z z = y)
13	12	19.32 1875	. . 3 ⊢ (∀z((∀z z = x ∨ ∀z z = y) ∨ (x = y → ∀z x = y)) ↔ ((∀z z = x ∨ ∀z z = y) ∨ ∀z(x = y → ∀z x = y)))
14	9, 13	mpbi 199	. 2 ⊢ ((∀z z = x ∨ ∀z z = y) ∨ ∀z(x = y → ∀z x = y))
15		orass 510	. 2 ⊢ (((∀z z = x ∨ ∀z z = y) ∨ ∀z(x = y → ∀z x = y)) ↔ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y))))
16	14, 15	mpbi 199	1 ⊢ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357  ∀wal 1540   = wceq 1642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)