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Theorem sylan2br 462

Description: A syllogism inference. (Contributed by NM, 21-Apr-1994.)

**Hypotheses**
Ref	Expression
sylan2br.1	⊢ (χ ↔ φ)
sylan2br.2	⊢ ((ψ ∧ χ) → θ)

**Assertion**
Ref	Expression
sylan2br	⊢ ((ψ ∧ φ) → θ)

**Proof of Theorem sylan2br**
Step	Hyp	Ref	Expression
1		sylan2br.1	. . 3 ⊢ (χ ↔ φ)
2	1	biimpri 197	. 2 ⊢ (φ → χ)
3		sylan2br.2	. 2 ⊢ ((ψ ∧ χ) → θ)
4	2, 3	sylan2 460	1 ⊢ ((ψ ∧ φ) → θ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8

This theorem depends on definitions: df-bi 177 df-an 360

This theorem is referenced by: syl2anbr 466 imainss 5042 xpexr2 5110 funeu2 5132 imadif 5171 fnop 5186 fcnvres 5243 fnopovb 5557 fovrn 5604 fnovrn 5607 nchoicelem17 6305