Theorem ralf0 2363

Description: The quantification of a falsehood is vacuous when true.

Hypothesis

Ref	Expression
ralf0.1	|- -. ph

Assertion

Ref	Expression
ralf0	|- (A.x e. A ph <-> A = (/))

Distinct variable group:   x,A

Proof of Theorem ralf0

Step	Hyp	Ref	Expression
1		ralf0.1	. . . . 5 |- -. ph
2		con3 94	. . . . 5 |- ((x e. A -> ph) -> (-. ph -> -. x e. A))
3	1, 2	mpi 44	. . . 4 |- ((x e. A -> ph) -> -. x e. A)
4	3	19.20i 994	. . 3 |- (A.x(x e. A -> ph) -> A.x -. x e. A)
5		df-ral 1652	. . 3 |- (A.x e. A ph <-> A.x(x e. A -> ph))
6		eq0 2298	. . 3 |- (A = (/) <-> A.x -. x e. A)
7	4, 5, 6	3imtr4 219	. 2 |- (A.x e. A ph -> A = (/))
8		rzal 2359	. 2 |- (A = (/) -> A.x e. A ph)
9	7, 8	impbi 157	1 |- (A.x e. A ph <-> A = (/))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146  A.wal 956   = wceq 958   e. wcel 960  A.wral 1648  (/)c0 2283

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-nul 2284

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