Theorem biantrud 545

Description: A wff is equivalent to its conjunction with truth.

Hypothesis

Ref	Expression
biantrud.1	|- (ph -> ps)

Assertion

Ref	Expression
biantrud	|- (ph -> (ch <-> (ch /\ ps)))

Proof of Theorem biantrud

Step	Hyp	Ref	Expression
1		biantrud.1	. . . . 5 |- (ph -> ps)
2	1	anim2i 270	. . . 4 |- ((ch /\ ph) -> (ch /\ ps))
3	2	exp 291	. . 3 |- (ch -> (ph -> (ch /\ ps)))
4	3	com12 13	. 2 |- (ph -> (ch -> (ch /\ ps)))
5		pm3.26 256	. . 3 |- ((ch /\ ps) -> ch)
6	5	a1i 7	. 2 |- (ph -> ((ch /\ ps) -> ch))
7	4, 6	impbid 397	1 |- (ph -> (ch <-> (ch /\ ps)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196

This theorem is referenced by:  biantrurd 546  mapdom2 3389  cardval 3633  nn2get 4438  shle0t 5367

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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