Theorem ad4ant23 1297

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

Hypothesis

Ref	Expression
ad4ant23.1	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
ad4ant23	⊢ ((((𝜃 ∧ 𝜑) ∧ 𝜓) ∧ 𝜏) → 𝜒)

Proof of Theorem ad4ant23

Step	Hyp	Ref	Expression
1		ad4ant23.1	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2	1	ex 450	. . . 4 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	a1dd 50	. . 3 ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜒)))
4	3	a1i 11	. 2 ⊢ (𝜃 → (𝜑 → (𝜓 → (𝜏 → 𝜒))))
5	4	imp41 619	1 ⊢ ((((𝜃 ∧ 𝜑) ∧ 𝜓) ∧ 𝜏) → 𝜒)

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:  fntpb  6473  usgredg2vlem2  26118  umgr3v3e3cycl  27044  matunitlindflem1  33405  matunitlindflem2  33406  heicant  33444  difmap  39399  xlimmnfvlem2  40059  xlimpnfvlem2  40063  sge0resplit  40623  hoidmvle  40814