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  962  mpteq12f  4078  opelopabsb  4254  elreimasng  4987  fvelrnb  5555  fmptco  5674  fconstfvm  5726  f1oiso  5817  canth  5819  mpoeq123  5924  dfoprab4f  6184  fmpox  6191  nnmword  6509  elfi  6960  ltmpig  7313  mul0eqap  8600  qreccl  9615  0fz1  10015  zmodid2  10322  divgcdcoprm0  12068  cnptoprest  13319  txrest  13356  cbvrald  14109