Theorem sylan9bbr 463

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 462	. 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜏))
4	3	ancoms 268	1 ⊢ ((𝜃 ∧ 𝜑) → (𝜓 ↔ 𝜏))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm5.75  964  mpteq12f  4105  opelopabsb  4285  elrelimasn  5019  fvelrnb  5591  fmptco  5710  fconstfvm  5762  f1oiso  5855  canth  5857  mpoeq123  5963  dfoprab4f  6226  fmpox  6233  nnmword  6551  elfi  7009  ltmpig  7378  mul0eqap  8668  qreccl  9684  0fz1  10087  zmodid2  10395  divgcdcoprm0  12150  cnptoprest  14260  txrest  14297  cbvrald  15079