Theorem bi2bian9 641

Description: Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.)

Hypotheses

Ref	Expression
bi2an9.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
bi2an9.2	⊢ (𝜃 → (𝜏 ↔ 𝜂))

Assertion

Ref	Expression
bi2bian9	⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ↔ 𝜏) ↔ (𝜒 ↔ 𝜂)))

Proof of Theorem bi2bian9

Step	Hyp	Ref	Expression
1		bi2an9.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	adantr 484	. 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜒))
3		bi2an9.2	. . 3 ⊢ (𝜃 → (𝜏 ↔ 𝜂))
4	3	adantl 485	. 2 ⊢ ((𝜑 ∧ 𝜃) → (𝜏 ↔ 𝜂))
5	2, 4	bibi12d 349	1 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ↔ 𝜏) ↔ (𝜒 ↔ 𝜂)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 400

This theorem is referenced by:  releccnveq  36040  extssr  36270  wepwsolem  40459  aomclem8  40478