Theorem nabi1 14139

Description: Constructor theorem for -/\.

Assertion

Ref	Expression
nabi1	|- ((ph <-> ps) -> ((ph -/\ ch) <-> (ps -/\ ch)))

Proof of Theorem nabi1

Step	Hyp	Ref	Expression
1		anbi1 683	. 2 |- ((ph <-> ps) -> ((ph /\ ch) <-> (ps /\ ch)))
2		notbi 581	. . 3 |- (((ph /\ ch) <-> (ps /\ ch)) <-> (-. (ph /\ ch) <-> -. (ps /\ ch)))
3	2	biimpi 168	. 2 |- (((ph /\ ch) <-> (ps /\ ch)) -> (-. (ph /\ ch) <-> -. (ps /\ ch)))
4		df-nand 1230	. . . 4 |- ((ph -/\ ch) <-> -. (ph /\ ch))
5		df-nand 1230	. . . 4 |- ((ps -/\ ch) <-> -. (ps /\ ch))
6	4, 5	bibi12i 672	. . 3 |- (((ph -/\ ch) <-> (ps -/\ ch)) <-> (-. (ph /\ ch) <-> -. (ps /\ ch)))
7	6	biimpri 169	. 2 |- ((-. (ph /\ ch) <-> -. (ps /\ ch)) -> ((ph -/\ ch) <-> (ps -/\ ch)))
8	1, 3, 7	3syl 24	1 |- ((ph <-> ps) -> ((ph -/\ ch) <-> (ps -/\ ch)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   -/\ wnand 1229

This theorem is referenced by:  nabi1i 14141

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

This theorem depends on definitions:  df-bi 164  df-an 242  df-nand 1230

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