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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)

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: syl2anb 465 anabsan 786 eqtr2 2371 pm13.181 2589 rmob 3134 sspsstr 3374 disjne 3596 xpcan2 5058 fssres 5238 funbrfvb 5360 fvco 5383 fvimacnvi 5402 ffvresb 5431 leaddc2 6215 lemuc2 6254