Theorem rzal 3934

Description: Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
rzal	|- ( A = (/) -> A. x e. A ph )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem rzal

Step	Hyp	Ref	Expression
1		ne0i 3799	. . . 4 |- ( x e. A -> A =/= (/) )
2	1	necon2bi 2694	. . 3 |- ( A = (/) -> -. x e. A )
3	2	pm2.21d 106	. 2 |- ( A = (/) -> ( x e. A -> ph ) )
4	3	ralrimiv 2869	1 |- ( A = (/) -> A. x e. A ph )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 1819   A.wral 2807   (/)c0 3793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-v 3111  df-dif 3474  df-nul 3794

This theorem is referenced by:  ralidm  3936  rgenz  3938  ralf0  3939  raaan  3940  raaanv  3941  iinrab2  4395  riinrab  4408  reusv2lem2  4658  cnvpo  5551  dffi3  7909  brdom3  8923  dedekind  9761  fimaxre2  10511  efgs1  16880  opnnei  19748  axcontlem12  24405  ubthlem1  25913  nofulllem2  29680  mbfresfi  30266  bddiblnc  30290  blbnd  30488  rrnequiv  30536  upbdrech2  31711  stoweidlem9  31994  fourierdlem31  32123  raaan2  32383