Theorem syl22anbrc 33049

Description: Syllogism inference. (Contributed by Thierry Arnoux, 19-Oct-2025.)

Hypotheses

Ref	Expression
syl22anbrc.1	⊢ (𝜑 → 𝜓)
syl22anbrc.2	⊢ (𝜑 → 𝜒)
syl22anbrc.3	⊢ (𝜑 → 𝜃)
syl22anbrc.4	⊢ (𝜑 → 𝜏)
syl22anbrc.5	⊢ (𝜂 ↔ ((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)))

Assertion

Ref	Expression
syl22anbrc	⊢ (𝜑 → 𝜂)

Proof of Theorem syl22anbrc

Step	Hyp	Ref	Expression
1		syl22anbrc.1	. 2 ⊢ (𝜑 → 𝜓)
2		syl22anbrc.2	. 2 ⊢ (𝜑 → 𝜒)
3		syl22anbrc.3	. . 3 ⊢ (𝜑 → 𝜃)
4		syl22anbrc.4	. . 3 ⊢ (𝜑 → 𝜏)
5	3, 4	jca 521	. 2 ⊢ (𝜑 → (𝜃 ∧ 𝜏))
6		syl22anbrc.5	. 2 ⊢ (𝜂 ↔ ((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)))
7	1, 2, 5, 6	syl21anbrc 1363	1 ⊢ (𝜑 → 𝜂)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  conjga  33724  mplvrpmga  34170  fldextrspundgdvdslem  34305  fldextrspundgdvds  34306