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  970  mpteq12f  4169  opelopabsb  4354  elrelimasn  5102  fvelrnb  5693  fmptco  5813  fconstfvm  5871  f1oiso  5966  canth  5968  mpoeq123  6079  elovmporab  6221  elovmporab1w  6222  dfoprab4f  6355  fmpox  6364  nnmword  6685  elfi  7169  ltmpig  7558  mul0eqap  8849  qreccl  9875  0fz1  10279  zmodid2  10613  ccatrcl1  11190  divgcdcoprm0  12672  cnptoprest  14962  txrest  14999  uhgreq12g  15926  cbvrald  16384