Theorem ad5ant24 1305

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

Hypothesis

Ref	Expression
ad5ant24.1	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
ad5ant24	⊢ (((((𝜃 ∧ 𝜑) ∧ 𝜏) ∧ 𝜓) ∧ 𝜂) → 𝜒)

Proof of Theorem ad5ant24

Step	Hyp	Ref	Expression
1		ad5ant24.1	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2	1	ex 450	. . . . . . . 8 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	2a1dd 51	. . . . . . 7 ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜏 → 𝜒))))
4	3	a1ddd 80	. . . . . 6 ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜂 → (𝜏 → 𝜒)))))
5	4	com45 97	. . . . 5 ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜏 → (𝜂 → 𝜒)))))
6	5	com3r 87	. . . 4 ⊢ (𝜃 → (𝜑 → (𝜓 → (𝜏 → (𝜂 → 𝜒)))))
7	6	com34 91	. . 3 ⊢ (𝜃 → (𝜑 → (𝜏 → (𝜓 → (𝜂 → 𝜒)))))
8	7	imp 445	. 2 ⊢ ((𝜃 ∧ 𝜑) → (𝜏 → (𝜓 → (𝜂 → 𝜒))))
9	8	imp41 619	1 ⊢ (((((𝜃 ∧ 𝜑) ∧ 𝜏) ∧ 𝜓) ∧ 𝜂) → 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:  metust  22363  clwlkclwwlklem2a4  26898  matunitlindflem1  33405  rexabslelem  39645