Theorem rzal 3377

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 3292	. . . 4 |- ( x e. A -> A =/= (/) )
2	1	necon2bi 2310	. . 3 |- ( A = (/) -> -. x e. A )
3	2	pm2.21d 584	. 2 |- ( A = (/) -> ( x e. A -> ph ) )
4	3	ralrimiv 2445	1 |- ( A = (/) -> A. x e. A ph )

Syntax hints:   -> wi 4   = wceq 1289   e. wcel 1438   A.wral 2359   (/)c0 3286

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 3001  df-nul 3287

This theorem is referenced by:  ralf0  3383