Theorem ad5ant14 1302

Description: Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Hypothesis

Ref	Expression
ad5ant14.1	|- ( ( ph /\ ps ) -> ch )

Assertion

Ref	Expression
ad5ant14	|- ( ( ( ( ( ph /\ th ) /\ ta ) /\ ps ) /\ et ) -> ch )

Proof of Theorem ad5ant14

Step	Hyp	Ref	Expression
1		ad5ant14.1	. . . . . . . . 9 |- ( ( ph /\ ps ) -> ch )
2	1	ex 450	. . . . . . . 8 |- ( ph -> ( ps -> ch ) )
3	2	2a1dd 51	. . . . . . 7 |- ( ph -> ( ps -> ( th -> ( ta -> ch ) ) ) )
4	3	a1ddd 80	. . . . . 6 |- ( ph -> ( ps -> ( th -> ( et -> ( ta -> ch ) ) ) ) )
5	4	com45 97	. . . . 5 |- ( ph -> ( ps -> ( th -> ( ta -> ( et -> ch ) ) ) ) )
6	5	com23 86	. . . 4 |- ( ph -> ( th -> ( ps -> ( ta -> ( et -> ch ) ) ) ) )
7	6	com34 91	. . 3 |- ( ph -> ( th -> ( ta -> ( ps -> ( et -> ch ) ) ) ) )
8	7	imp 445	. 2 |- ( ( ph /\ th ) -> ( ta -> ( ps -> ( et -> ch ) ) ) )
9	8	imp41 619	1 |- ( ( ( ( ( ph /\ th ) /\ ta ) /\ ps ) /\ et ) -> ch )

Syntax hints:   -> wi 4   /\ wa 384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-an 386

This theorem is referenced by:  legval  25479  lindsenlbs  33404  matunitlindflem1  33405  xrralrecnnle  39602  pimdecfgtioo  40927  pimincfltioo  40928