Theorem r19.27av 1754

Description: Restricted version of one direction of Theorem 19.27 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.)

Assertion

Ref	Expression
r19.27av	|- ((A.x e. A ph /\ ps) -> A.x e. A (ph /\ ps))

Distinct variable group:   ps,x

Proof of Theorem r19.27av

Step	Hyp	Ref	Expression
1		pm2.27 62	. . . . 5 |- (x e. A -> ((x e. A -> ph) -> ph))
2	1	anim1d 560	. . . 4 |- (x e. A -> (((x e. A -> ph) /\ ps) -> (ph /\ ps)))
3	2	com12 11	. . 3 |- (((x e. A -> ph) /\ ps) -> (x e. A -> (ph /\ ps)))
4	3	19.20i 992	. 2 |- (A.x((x e. A -> ph) /\ ps) -> A.x(x e. A -> (ph /\ ps)))
5		df-ral 1649	. . . 4 |- (A.x e. A ph <-> A.x(x e. A -> ph))
6	5	anbi1i 481	. . 3 |- ((A.x e. A ph /\ ps) <-> (A.x(x e. A -> ph) /\ ps))
7		19.27v 1298	. . 3 |- (A.x((x e. A -> ph) /\ ps) <-> (A.x(x e. A -> ph) /\ ps))
8	6, 7	bitr4 176	. 2 |- ((A.x e. A ph /\ ps) <-> A.x((x e. A -> ph) /\ ps))
9		df-ral 1649	. 2 |- (A.x e. A (ph /\ ps) <-> A.x(x e. A -> (ph /\ ps)))
10	4, 8, 9	3imtr4 219	1 |- ((A.x e. A ph /\ ps) -> A.x e. A (ph /\ ps))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   e. wcel 958  A.wral 1645

This theorem is referenced by:  r19.28av 1755  spanun 9467

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1649

Copyright terms: Public domain