Theorem axi12 2600

Description: Axiom of Quantifier Introduction (intuitionistic logic axiom ax-i12). In classical logic, this is mostly a restatement of axc9 2302 (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	|- ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )

Proof of Theorem axi12

Step	Hyp	Ref	Expression
1		nfae 2316	. . . . 5 |- F/ z A. z z = x
2		nfae 2316	. . . . 5 |- F/ z A. z z = y
3	1, 2	nfor 1834	. . . 4 |- F/ z ( A. z z = x \/ A. z z = y )
4	3	19.32 2101	. . 3 |- ( A. z ( ( A. z z = x \/ A. z z = y ) \/ ( x = y -> A. z x = y ) ) <-> ( ( A. z z = x \/ A. z z = y ) \/ A. z ( x = y -> A. z x = y ) ) )
5		axc9 2302	. . . . . 6 |- ( -. A. z z = x -> ( -. A. z z = y -> ( x = y -> A. z x = y ) ) )
6	5	orrd 393	. . . . 5 |- ( -. A. z z = x -> ( A. z z = y \/ ( x = y -> A. z x = y ) ) )
7	6	orri 391	. . . 4 |- ( A. z z = x \/ ( A. z z = y \/ ( x = y -> A. z x = y ) ) )
8		orass 546	. . . 4 |- ( ( ( A. z z = x \/ A. z z = y ) \/ ( x = y -> A. z x = y ) ) <-> ( A. z z = x \/ ( A. z z = y \/ ( x = y -> A. z x = y ) ) ) )
9	7, 8	mpbir 221	. . 3 |- ( ( A. z z = x \/ A. z z = y ) \/ ( x = y -> A. z x = y ) )
10	4, 9	mpgbi 1725	. 2 |- ( ( A. z z = x \/ A. z z = y ) \/ A. z ( x = y -> A. z x = y ) )
11		orass 546	. 2 |- ( ( ( A. z z = x \/ A. z z = y ) \/ A. z ( x = y -> A. z x = y ) ) <-> ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) ) )
12	10, 11	mpbi 220	1 |- ( A. z z = x \/ ( A. z z = y \/ A. z ( x = y -> A. z x = y ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)