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)

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

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