Theorem bnj1476 30917

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1476.1	|- D = { x e. A | -. ph }
bnj1476.2	|- ( ps -> D = (/) )

Assertion

Ref	Expression
bnj1476	|- ( ps -> A. x e. A ph )

Proof of Theorem bnj1476

Step	Hyp	Ref	Expression
1		bnj1476.2	. . . 4 |- ( ps -> D = (/) )
2		bnj1476.1	. . . . . 6 |- D = { x e. A | -. ph }
3		nfrab1 3122	. . . . . 6 |- F/_ x { x e. A | -. ph }
4	2, 3	nfcxfr 2762	. . . . 5 |- F/_ x D
5	4	eq0f 3925	. . . 4 |- ( D = (/) <-> A. x -. x e. D )
6	1, 5	sylib 208	. . 3 |- ( ps -> A. x -. x e. D )
7	2	rabeq2i 3197	. . . . 5 |- ( x e. D <-> ( x e. A /\ -. ph ) )
8	7	notbii 310	. . . 4 |- ( -. x e. D <-> -. ( x e. A /\ -. ph ) )
9		iman 440	. . . 4 |- ( ( x e. A -> ph ) <-> -. ( x e. A /\ -. ph ) )
10	8, 9	sylbb2 228	. . 3 |- ( -. x e. D -> ( x e. A -> ph ) )
11	6, 10	sylg 1750	. 2 |- ( ps -> A. x ( x e. A -> ph ) )
12	11	bnj1142 30860	1 |- ( ps -> A. x e. A ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  bnj1312  31126