Theorem ralnralall 4080

Description: A contradiction concerning restricted generalization for a nonempty set implies anything. (Contributed by Alexander van der Vekens, 4-Sep-2018.)

Assertion

Ref	Expression
ralnralall	|- ( A =/= (/) -> ( ( A. x e. A ph /\ A. x e. A -. ph ) -> ps ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem ralnralall

Step	Hyp	Ref	Expression
1		r19.26 3064	. 2 |- ( A. x e. A ( ph /\ -. ph ) <-> ( A. x e. A ph /\ A. x e. A -. ph ) )
2		pm3.24 926	. . . . 5 |- -. ( ph /\ -. ph )
3	2	bifal 1497	. . . 4 |- ( ( ph /\ -. ph ) <-> F. )
4	3	ralbii 2980	. . 3 |- ( A. x e. A ( ph /\ -. ph ) <-> A. x e. A F. )
5		r19.3rzv 4064	. . . 4 |- ( A =/= (/) -> ( F. <-> A. x e. A F. ) )
6		falim 1498	. . . 4 |- ( F. -> ps )
7	5, 6	syl6bir 244	. . 3 |- ( A =/= (/) -> ( A. x e. A F. -> ps ) )
8	4, 7	syl5bi 232	. 2 |- ( A =/= (/) -> ( A. x e. A ( ph /\ -. ph ) -> ps ) )
9	1, 8	syl5bir 233	1 |- ( A =/= (/) -> ( ( A. x e. A ph /\ A. x e. A -. ph ) -> ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   F. wfal 1488   =/= wne 2794   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-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)