Theorem ralf0 4078

Description: The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.)

Hypothesis

Ref	Expression
ralf0.1	|- -. ph

Assertion

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

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem ralf0

Step	Hyp	Ref	Expression
1		ralf0.1	. . . 4 |- -. ph
2		mtt 354	. . . 4 |- ( -. ph -> ( -. x e. A <-> ( x e. A -> ph ) ) )
3	1, 2	ax-mp 5	. . 3 |- ( -. x e. A <-> ( x e. A -> ph ) )
4	3	albii 1747	. 2 |- ( A. x -. x e. A <-> A. x ( x e. A -> ph ) )
5		eq0 3929	. 2 |- ( A = (/) <-> A. x -. x e. A )
6		df-ral 2917	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
7	4, 5, 6	3bitr4ri 293	1 |- ( A. x e. A ph <-> A = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)