Theorem sylan9bbr 681

Description: Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 4-Mar-1995.)

Hypotheses

Ref	Expression
sylan9bbr.1	⊢ (φ → (ψ ↔ χ))
sylan9bbr.2	⊢ (θ → (χ ↔ τ))

Assertion

Ref	Expression
sylan9bbr	⊢ ((θ ∧ φ) → (ψ ↔ τ))

Proof of Theorem sylan9bbr

Step	Hyp	Ref	Expression
1		sylan9bbr.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2		sylan9bbr.2	. . 3 ⊢ (θ → (χ ↔ τ))
3	1, 2	sylan9bb 680	. 2 ⊢ ((φ ∧ θ) → (ψ ↔ τ))
4	3	ancoms 439	1 ⊢ ((θ ∧ φ) → (ψ ↔ τ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

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

This theorem is used by:  pm5.75  903  sbcom  2089  sbcom2  2114  2mo  2282  2eu6  2289  elssetkg  4270  fconstfv  5457  f1oiso  5500  mpteq12f  5656  mpt2eq123  5662  fmpt2x  5731  sbth  6207