Theorem sylanbr 345

Description: A syllogism inference.

Hypotheses

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

Assertion

Ref	Expression
sylanbr	⊢ ((θ ∧ ψ) → χ)

Proof of Theorem sylanbr

Step	Hyp	Ref	Expression
1		sylan.1	. 2 ⊢ ((φ ∧ ψ) → χ)
2		sylanbr.2	. . 3 ⊢ (φ ↔ θ)
3	2	biimpr 134	. 2 ⊢ (θ → φ)
4	1, 3	sylan 343	1 ⊢ ((θ ∧ ψ) → χ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196

This theorem is referenced by:  syl2anbr 351  funfvop 2857  funfv 2862  fvopab2 2878  th3qlem1 3250  axrnegex 4080

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

metamath.org