Theorem ad5ant15 1303

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

Hypothesis

Ref	Expression
ad5ant15.1	|- ( ( ph /\ ps ) -> ch )

Assertion

Ref	Expression
ad5ant15	|- ( ( ( ( ( ph /\ th ) /\ ta ) /\ et ) /\ ps ) -> ch )

Proof of Theorem ad5ant15

Step	Hyp	Ref	Expression
1		ad5ant15.1	. . . . . . . . . 10 |- ( ( ph /\ ps ) -> ch )
2	1	ex 450	. . . . . . . . 9 |- ( ph -> ( ps -> ch ) )
3	2	2a1dd 51	. . . . . . . 8 |- ( ph -> ( ps -> ( th -> ( ta -> ch ) ) ) )
4	3	a1ddd 80	. . . . . . 7 |- ( ph -> ( ps -> ( th -> ( et -> ( ta -> ch ) ) ) ) )
5	4	com45 97	. . . . . 6 |- ( ph -> ( ps -> ( th -> ( ta -> ( et -> ch ) ) ) ) )
6	5	com23 86	. . . . 5 |- ( ph -> ( th -> ( ps -> ( ta -> ( et -> ch ) ) ) ) )
7	6	com34 91	. . . 4 |- ( ph -> ( th -> ( ta -> ( ps -> ( et -> ch ) ) ) ) )
8	7	com45 97	. . 3 |- ( ph -> ( th -> ( ta -> ( et -> ( ps -> ch ) ) ) ) )
9	8	imp 445	. 2 |- ( ( ph /\ th ) -> ( ta -> ( et -> ( ps -> ch ) ) ) )
10	9	imp41 619	1 |- ( ( ( ( ( ph /\ th ) /\ ta ) /\ et ) /\ ps ) -> 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:  breprexplemc  30710  mblfinlem2  33447  supxrgelem  39553  supxrge  39554  rexabslelem  39645  uzub  39658  smflimlem4  40982