Theorem axi12 2433

Description: Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12). In classical logic, this is mostly a restatement of axc9 2046 (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

Proof of Theorem axi12

Step	Hyp	Ref	Expression
1		nfae 2056	. . . . 5
2		nfae 2056	. . . . 5
3	1, 2	nfor 1935	. . . 4
4	3	19.32 1967	. . 3
5		axc9 2046	. . . . . 6
6	5	orrd 378	. . . . 5
7	6	orri 376	. . . 4
8		orass 524	. . . 4
9	7, 8	mpbir 209	. . 3
10	4, 9	mpgbi 1621	. 2
11		orass 524	. 2
12	10, 11	mpbi 208	1

Syntax hints:  -.wn 3  ->wi 4  \/wo 368  A.wal 1393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1613  df-nf 1617