Theorem 19.26-3an 1595

Description: Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.)

Assertion

Ref	Expression
19.26-3an	|- (A.x(ph /\ ps /\ ch) <-> (A.xph /\ A.xps /\ A.xch))

Proof of Theorem 19.26-3an

Step	Hyp	Ref	Expression
1		19.26 1593	. . 3 |- (A.x((ph /\ ps) /\ ch) <-> (A.x(ph /\ ps) /\ A.xch))
2		19.26 1593	. . . 4 |- (A.x(ph /\ ps) <-> (A.xph /\ A.xps))
3	2	anbi1i 676	. . 3 |- ((A.x(ph /\ ps) /\ A.xch) <-> ((A.xph /\ A.xps) /\ A.xch))
4	1, 3	bitri 240	. 2 |- (A.x((ph /\ ps) /\ ch) <-> ((A.xph /\ A.xps) /\ A.xch))
5		df-3an 936	. . 3 |- ((ph /\ ps /\ ch) <-> ((ph /\ ps) /\ ch))
6	5	albii 1566	. 2 |- (A.x(ph /\ ps /\ ch) <-> A.x((ph /\ ps) /\ ch))
7		df-3an 936	. 2 |- ((A.xph /\ A.xps /\ A.xch) <-> ((A.xph /\ A.xps) /\ A.xch))
8	4, 6, 7	3bitr4i 268	1 |- (A.x(ph /\ ps /\ ch) <-> (A.xph /\ A.xps /\ A.xch))

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934  A.wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is referenced by: (None)