Theorem syl2an3an 1339

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

Hypotheses

Ref	Expression
syl2an3an.1	⊢ (𝜑 → 𝜓)
syl2an3an.2	⊢ (𝜑 → 𝜒)
syl2an3an.3	⊢ (𝜃 → 𝜏)
syl2an3an.4	⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
syl2an3an	⊢ ((𝜑 ∧ 𝜃) → 𝜂)

Proof of Theorem syl2an3an

Step	Hyp	Ref	Expression
1		syl2an3an.1	. . 3 ⊢ (𝜑 → 𝜓)
2		syl2an3an.2	. . 3 ⊢ (𝜑 → 𝜒)
3		syl2an3an.3	. . 3 ⊢ (𝜃 → 𝜏)
4		syl2an3an.4	. . 3 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) → 𝜂)
5	1, 2, 3, 4	syl3an 1320	. 2 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜃) → 𝜂)
6	5	3anidm12 1336	1 ⊢ ((𝜑 ∧ 𝜃) → 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:  prfidceq  7235  ccatass  11392  ccatpfx  11489  swrdswrd  11493  expcnvap0  12288  efexp  12468  cncongr1  12900  uptx  15466  logbgcd1irr  16169  bpos  16286  gausslemma2dlem2  16352  wlkeq  16766