Theorem rzal 3520

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 3429	. . . 4 |- ( x e. A -> A =/= (/) )
2	1	necon2bi 2402	. . 3 |- ( A = (/) -> -. x e. A )
3	2	pm2.21d 619	. 2 |- ( A = (/) -> ( x e. A -> ph ) )
4	3	ralrimiv 2549	1 |- ( A = (/) -> A. x e. A ph )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   A.wral 2455   (/)c0 3422

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-v 2739  df-dif 3131  df-nul 3423

This theorem is referenced by:  ralf0  3526  fiubm  10800  mgm0  12719  sgrp0  12746