Theorem ralralimp 41295

Description: Selecting one of two alternatives within a restricted generalization if one of the alternatives is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.)

Assertion

Ref	Expression
ralralimp	|- ( ( ph /\ A =/= (/) ) -> ( A. x e. A ( ( ph -> ( th \/ ta ) ) /\ -. th ) -> ta ) )

Distinct variable groups:   x, A   ph, x   ta, x

Allowed substitution hint:   th( x)

Proof of Theorem ralralimp

Step	Hyp	Ref	Expression
1		ornld 940	. . . 4 |- ( ph -> ( ( ( ph -> ( th \/ ta ) ) /\ -. th ) -> ta ) )
2	1	adantr 481	. . 3 |- ( ( ph /\ A =/= (/) ) -> ( ( ( ph -> ( th \/ ta ) ) /\ -. th ) -> ta ) )
3	2	ralimdv 2963	. 2 |- ( ( ph /\ A =/= (/) ) -> ( A. x e. A ( ( ph -> ( th \/ ta ) ) /\ -. th ) -> A. x e. A ta ) )
4		rspn0 3934	. . 3 |- ( A =/= (/) -> ( A. x e. A ta -> ta ) )
5	4	adantl 482	. 2 |- ( ( ph /\ A =/= (/) ) -> ( A. x e. A ta -> ta ) )
6	3, 5	syld 47	1 |- ( ( ph /\ A =/= (/) ) -> ( A. x e. A ( ( ph -> ( th \/ ta ) ) /\ -. th ) -> ta ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   =/= 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-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)