Theorem bi2bian9 480

Description: Deduction joining two biconditionals with different antecedents.

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 306	. 2 ⊢ ((φ ∧ θ) → (ψ ↔ χ))
3		bi2an9.2	. . 3 ⊢ (θ → (τ ↔ η))
4	3	adantl 305	. 2 ⊢ ((φ ∧ θ) → (τ ↔ η))
5	2, 4	bibi12d 477	1 ⊢ ((φ ∧ θ) → ((ψ ↔ τ) ↔ (χ ↔ η)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196

This theorem is referenced by:  uzind 4603

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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