Theorem exlimddvfi 33927

Description: A lemma for eliminating an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.)

Hypotheses

Ref	Expression
exlimddvfi.1	|- ( ph -> E. x th )
exlimddvfi.2	|- F/ y th
exlimddvfi.3	|- F/ y ps
exlimddvfi.4	|- ( [. y / x ]. th <-> et )
exlimddvfi.5	|- ( ( et /\ ps ) -> ch )
exlimddvfi.6	|- F/ y ch

Assertion

Ref	Expression
exlimddvfi	|- ( ( ph /\ ps ) -> ch )

Proof of Theorem exlimddvfi

Step	Hyp	Ref	Expression
1		exlimddvfi.1	. . 3 |- ( ph -> E. x th )
2		exlimddvfi.2	. . . 4 |- F/ y th
3	2	sb8e 2425	. . 3 |- ( E. x th <-> E. y [ y / x ] th )
4	1, 3	sylib 208	. 2 |- ( ph -> E. y [ y / x ] th )
5		exlimddvfi.3	. 2 |- F/ y ps
6		sbsbc 3439	. . . 4 |- ( [ y / x ] th <-> [. y / x ]. th )
7		exlimddvfi.4	. . . 4 |- ( [. y / x ]. th <-> et )
8	6, 7	bitri 264	. . 3 |- ( [ y / x ] th <-> et )
9		exlimddvfi.5	. . 3 |- ( ( et /\ ps ) -> ch )
10	8, 9	sylanb 489	. 2 |- ( ( [ y / x ] th /\ ps ) -> ch )
11		exlimddvfi.6	. 2 |- F/ y ch
12	4, 5, 10, 11	exlimddvf 33926	1 |- ( ( ph /\ ps ) -> ch )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   E.wex 1704   F/wnf 1708   [wsb 1880   [.wsbc 3435

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-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-sbc 3436

This theorem is referenced by: (None)