Theorem r19.36zv 3650

Description: Restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 20-Sep-2003.)

Assertion

Ref	Expression
r19.36zv	|- (A =/= (/) -> (E.x e. A (ph -> ps) <-> (A.x e. A ph -> ps)))

Distinct variable groups:   x,A   ps,x

Allowed substitution hint:   ph(x)

Proof of Theorem r19.36zv

Step	Hyp	Ref	Expression
1		r19.9rzv 3644	. . 3 |- (A =/= (/) -> (ps <-> E.x e. A ps))
2	1	imbi2d 307	. 2 |- (A =/= (/) -> ((A.x e. A ph -> ps) <-> (A.x e. A ph -> E.x e. A ps)))
3		r19.35 2758	. 2 |- (E.x e. A (ph -> ps) <-> (A.x e. A ph -> E.x e. A ps))
4	2, 3	syl6rbbr 255	1 |- (A =/= (/) -> (E.x e. A (ph -> ps) <-> (A.x e. A ph -> ps)))

Syntax hints:   -> wi 4   <-> wb 176   =/= wne 2516  A.wral 2614  E.wrex 2615  (/)c0 3550

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551

This theorem is referenced by: (None)