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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 = yz x = y)))

Proof of Theorem axi12
StepHypRef Expression
1 ax12o 1934 . . . . . . 7 z z = x → (¬ z z = y → (x = yz x = y)))
2 df-or 359 . . . . . . . 8 ((z z = y (x = yz x = y)) ↔ (¬ z z = y → (x = yz x = y)))
32imbi2i 303 . . . . . . 7 ((¬ z z = x → (z z = y (x = yz x = y))) ↔ (¬ z z = x → (¬ z z = y → (x = yz x = y))))
41, 3mpbir 200 . . . . . 6 z z = x → (z z = y (x = yz x = y)))
5 df-or 359 . . . . . 6 ((z z = x (z z = y (x = yz x = y))) ↔ (¬ z z = x → (z z = y (x = yz x = y))))
64, 5mpbir 200 . . . . 5 (z z = x (z z = y (x = yz x = y)))
7 orass 510 . . . . 5 (((z z = x z z = y) (x = yz x = y)) ↔ (z z = x (z z = y (x = yz x = y))))
86, 7mpbir 200 . . . 4 ((z z = x z z = y) (x = yz x = y))
98ax-gen 1546 . . 3 z((z z = x z z = y) (x = yz x = y))
10 nfa1 1788 . . . . 5 zz z = x
11 nfa1 1788 . . . . 5 zz z = y
1210, 11nfor 1836 . . . 4 z(z z = x z z = y)
131219.32 1875 . . 3 (z((z z = x z z = y) (x = yz x = y)) ↔ ((z z = x z z = y) z(x = yz x = y)))
149, 13mpbi 199 . 2 ((z z = x z z = y) z(x = yz x = y))
15 orass 510 . 2 (((z z = x z z = y) z(x = yz x = y)) ↔ (z z = x (z z = y z(x = yz x = y))))
1614, 15mpbi 199 1 (z z = x (z z = y z(x = yz x = y)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357  ∀wal 1540   = wceq 1642 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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)
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