Theorem syl2an23an 1340

Description: Deduction related to syl3an 1320 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)

Hypotheses

Ref	Expression
syl2an23an.1	⊢ (𝜑 → 𝜓)
syl2an23an.2	⊢ (𝜑 → 𝜒)
syl2an23an.3	⊢ ((𝜃 ∧ 𝜑) → 𝜏)
syl2an23an.4	⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
syl2an23an	⊢ ((𝜃 ∧ 𝜑) → 𝜂)

Proof of Theorem syl2an23an

Step	Hyp	Ref	Expression
1		syl2an23an.3	. . 3 ⊢ ((𝜃 ∧ 𝜑) → 𝜏)
2		syl2an23an.1	. . . 4 ⊢ (𝜑 → 𝜓)
3		syl2an23an.2	. . . 4 ⊢ (𝜑 → 𝜒)
4		syl2an23an.4	. . . . 5 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) → 𝜂)
5	4	3exp 1233	. . . 4 ⊢ (𝜓 → (𝜒 → (𝜏 → 𝜂)))
6	2, 3, 5	sylc 62	. . 3 ⊢ (𝜑 → (𝜏 → 𝜂))
7	1, 6	syl5 32	. 2 ⊢ (𝜑 → ((𝜃 ∧ 𝜑) → 𝜂))
8	7	anabsi7 587	1 ⊢ ((𝜃 ∧ 𝜑) → 𝜂)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∧ w3a 1009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  swrdnd  11447  fsum3ser  12183  pcz  13134  fldivp1  13150  umgrvad2edg  16623