Theorem ralf0 2330

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 968	. . 3 |- (A.x(x e. A -> ph) -> A.x -. x e. A)
5		df-ral 1625	. . 3 |- (A.x e. A ph <-> A.x(x e. A -> ph))
6		eq0 2265	. . 3 |- (A = (/) <-> A.x -. x e. A)
7	4, 5, 6	3imtr4 219	. 2 |- (A.x e. A ph -> A = (/))
8		rzal 2326	. 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 950   = wceq 1099   e. wcel 1105  A.wral 1621  (/)c0 2251

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-v 1787  df-dif 2020  df-nul 2252

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