Theorem bnj115 30791

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj115.1	|- ( et <-> A. n e. D ( ta -> th ) )

Assertion

Ref	Expression
bnj115	|- ( et <-> A. n ( ( n e. D /\ ta ) -> th ) )

Proof of Theorem bnj115

Step	Hyp	Ref	Expression
1		bnj115.1	. 2 |- ( et <-> A. n e. D ( ta -> th ) )
2		df-ral 2917	. 2 |- ( A. n e. D ( ta -> th ) <-> A. n ( n e. D -> ( ta -> th ) ) )
3		impexp 462	. . . 4 |- ( ( ( n e. D /\ ta ) -> th ) <-> ( n e. D -> ( ta -> th ) ) )
4	3	bicomi 214	. . 3 |- ( ( n e. D -> ( ta -> th ) ) <-> ( ( n e. D /\ ta ) -> th ) )
5	4	albii 1747	. 2 |- ( A. n ( n e. D -> ( ta -> th ) ) <-> A. n ( ( n e. D /\ ta ) -> th ) )
6	1, 2, 5	3bitri 286	1 |- ( et <-> A. n ( ( n e. D /\ ta ) -> th ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ral 2917

This theorem is referenced by:  bnj953  31009  bnj964  31013  bnj1090  31047  bnj1112  31051