Theorem syl3anbrc 1136

Description: Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.)

Hypotheses

Ref	Expression
syl3anbrc.1	⊢ (φ → ψ)
syl3anbrc.2	⊢ (φ → χ)
syl3anbrc.3	⊢ (φ → θ)
syl3anbrc.4	⊢ (τ ↔ (ψ ∧ χ ∧ θ))

Assertion

Ref	Expression
syl3anbrc	⊢ (φ → τ)

Proof of Theorem syl3anbrc

Step	Hyp	Ref	Expression
1		syl3anbrc.1	. . 3 ⊢ (φ → ψ)
2		syl3anbrc.2	. . 3 ⊢ (φ → χ)
3		syl3anbrc.3	. . 3 ⊢ (φ → θ)
4	1, 2, 3	3jca 1132	. 2 ⊢ (φ → (ψ ∧ χ ∧ θ))
5		syl3anbrc.4	. 2 ⊢ (τ ↔ (ψ ∧ χ ∧ θ))
6	4, 5	sylibr 203	1 ⊢ (φ → τ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ w3a 934

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

This proof depends on definitions:  df-bi 177  df-an 360  df-3an 936

This theorem is used by:  sfindbl  4531  vfinspsslem1  4551  pod  5937  fundmen  6044  enmap1  6075  enprmap  6083