Theorem rzal 3535

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 3444	. . . 4 |- ( x e. A -> A =/= (/) )
2	1	necon2bi 2415	. . 3 |- ( A = (/) -> -. x e. A )
3	2	pm2.21d 620	. 2 |- ( A = (/) -> ( x e. A -> ph ) )
4	3	ralrimiv 2562	1 |- ( A = (/) -> A. x e. A ph )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2160   A.wral 2468   (/)c0 3437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-v 2754  df-dif 3146  df-nul 3438

This theorem is referenced by:  ralf0  3541  fiubm  10835  mgm0  12838  sgrp0  12866