Theorem sylanb 458

Description: A syllogism inference. (Contributed by NM, 18-May-1994.)

Hypotheses

Ref	Expression
sylanb.1	⊢ (φ ↔ ψ)
sylanb.2	⊢ ((ψ ∧ χ) → θ)

Assertion

Ref	Expression
sylanb	⊢ ((φ ∧ χ) → θ)

Proof of Theorem sylanb

Step	Hyp	Ref	Expression
1		sylanb.1	. . 3 ⊢ (φ ↔ ψ)
2	1	biimpi 186	. 2 ⊢ (φ → ψ)
3		sylanb.2	. 2 ⊢ ((ψ ∧ χ) → θ)
4	2, 3	sylan 457	1 ⊢ ((φ ∧ χ) → 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:  syl2anb  465  anabsan  786  eqtr2  2371  pm13.181  2590  rmob  3135  sspsstr  3375  disjne  3597  xpcan2  5059  fssres  5239  funbrfvb  5361  fvco  5384  fvimacnvi  5403  ffvresb  5432  leaddc2  6216  lemuc2  6255