Theorem r19.26-3 2749

Description: Theorem 19.26 of [Margaris] p. 90 with 3 restricted quantifiers. (Contributed by FL, 22-Nov-2010.)

Assertion

Ref	Expression
r19.26-3	|- (A.x e. A (ph /\ ps /\ ch) <-> (A.x e. A ph /\ A.x e. A ps /\ A.x e. A ch))

Proof of Theorem r19.26-3

Step	Hyp	Ref	Expression
1		df-3an 936	. . 3 |- ((ph /\ ps /\ ch) <-> ((ph /\ ps) /\ ch))
2	1	ralbii 2639	. 2 |- (A.x e. A (ph /\ ps /\ ch) <-> A.x e. A ((ph /\ ps) /\ ch))
3		r19.26 2747	. 2 |- (A.x e. A ((ph /\ ps) /\ ch) <-> (A.x e. A (ph /\ ps) /\ A.x e. A ch))
4		r19.26 2747	. . . 4 |- (A.x e. A (ph /\ ps) <-> (A.x e. A ph /\ A.x e. A ps))
5	4	anbi1i 676	. . 3 |- ((A.x e. A (ph /\ ps) /\ A.x e. A ch) <-> ((A.x e. A ph /\ A.x e. A ps) /\ A.x e. A ch))
6		df-3an 936	. . 3 |- ((A.x e. A ph /\ A.x e. A ps /\ A.x e. A ch) <-> ((A.x e. A ph /\ A.x e. A ps) /\ A.x e. A ch))
7	5, 6	bitr4i 243	. 2 |- ((A.x e. A (ph /\ ps) /\ A.x e. A ch) <-> (A.x e. A ph /\ A.x e. A ps /\ A.x e. A ch))
8	2, 3, 7	3bitri 262	1 |- (A.x e. A (ph /\ ps /\ ch) <-> (A.x e. A ph /\ A.x e. A ps /\ A.x e. A ch))

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by: (None)