Theorem rzal

Description: Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997) (Proof shortened by Andrew Salmon, 26-Jun-2011) Avoid df-clel , ax-8 . (Revised by Gino Giotto, 2-Sep-2024)

	Ref	Expression
Assertion	rzal	|- ( A = (/) -> A. x e. A ph )

Proof

Step	Hyp	Ref	Expression
1		dfcleq	|- ( A = { y | F. } <-> A. x ( x e. A <-> x e. { y | F. } ) )
2	1	biimpi	|- ( A = { y | F. } -> A. x ( x e. A <-> x e. { y | F. } ) )
3		df-clab	|- ( x e. { y | F. } <-> [ x / y ] F. )
4		sbv	|- ( [ x / y ] F. <-> F. )
5	3 4	bitri	|- ( x e. { y | F. } <-> F. )
6	5	bibi2i	|- ( ( x e. A <-> x e. { y | F. } ) <-> ( x e. A <-> F. ) )
7		nbfal	|- ( -. x e. A <-> ( x e. A <-> F. ) )
8		pm2.21	|- ( -. x e. A -> ( x e. A -> ph ) )
9	7 8	sylbir	|- ( ( x e. A <-> F. ) -> ( x e. A -> ph ) )
10	6 9	sylbi	|- ( ( x e. A <-> x e. { y | F. } ) -> ( x e. A -> ph ) )
11	2 10	sylg	|- ( A = { y | F. } -> A. x ( x e. A -> ph ) )
12		dfnul4	|- (/) = { y | F. }
13	12	eqeq2i	|- ( A = (/) <-> A = { y | F. } )
14		df-ral	|- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
15	11 13 14	3imtr4i	|- ( A = (/) -> A. x e. A ph )