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 GG, 2-Sep-2024)

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

Proof

Step	Hyp	Ref	Expression
1		biidd	|- ( y = x -> ( F. <-> F. ) )
2	1	eqabbw	|- ( A = { y | F. } <-> A. x ( x e. A <-> F. ) )
3	2	biimpi	|- ( A = { y | F. } -> A. x ( x e. A <-> F. ) )
4		nbfal	|- ( -. x e. A <-> ( x e. A <-> F. ) )
5		pm2.21	|- ( -. x e. A -> ( x e. A -> ph ) )
6	4 5	sylbir	|- ( ( x e. A <-> F. ) -> ( x e. A -> ph ) )
7	3 6	sylg	|- ( A = { y | F. } -> A. x ( x e. A -> ph ) )
8		dfnul4	|- (/) = { y | F. }
9	8	eqeq2i	|- ( A = (/) <-> A = { y | F. } )
10		df-ral	|- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
11	7 9 10	3imtr4i	|- ( A = (/) -> A. x e. A ph )