Theorem ad5ant135 1314

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

Hypothesis

Ref	Expression
ad5ant135.1	|- ( ( ph /\ ps /\ ch ) -> th )

Assertion

Ref	Expression
ad5ant135	|- ( ( ( ( ( ph /\ ta ) /\ ps ) /\ et ) /\ ch ) -> th )

Proof of Theorem ad5ant135

Step	Hyp	Ref	Expression
1		ad5ant135.1	. . . . . . . . . 10 |- ( ( ph /\ ps /\ ch ) -> th )
2	1	3exp 1264	. . . . . . . . 9 |- ( ph -> ( ps -> ( ch -> th ) ) )
3	2	a1ddd 80	. . . . . . . 8 |- ( ph -> ( ps -> ( ch -> ( ta -> th ) ) ) )
4	3	a1ddd 80	. . . . . . 7 |- ( ph -> ( ps -> ( ch -> ( et -> ( ta -> th ) ) ) ) )
5	4	com45 97	. . . . . 6 |- ( ph -> ( ps -> ( ch -> ( ta -> ( et -> th ) ) ) ) )
6	5	com34 91	. . . . 5 |- ( ph -> ( ps -> ( ta -> ( ch -> ( et -> th ) ) ) ) )
7	6	com23 86	. . . 4 |- ( ph -> ( ta -> ( ps -> ( ch -> ( et -> th ) ) ) ) )
8	7	com45 97	. . 3 |- ( ph -> ( ta -> ( ps -> ( et -> ( ch -> th ) ) ) ) )
9	8	imp 445	. 2 |- ( ( ph /\ ta ) -> ( ps -> ( et -> ( ch -> th ) ) ) )
10	9	imp41 619	1 |- ( ( ( ( ( ph /\ ta ) /\ ps ) /\ et ) /\ ch ) -> th )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037

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  df-3an 1039

This theorem is referenced by:  supxrgelem  39553  rexabslelem  39645  hoicvr  40762  hspmbllem2  40841  smfaddlem1  40971