Theorem bi2anan9r 479

Description: Deduction joining two equivalences to form equivalence of conjunctions.

Hypotheses

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

Assertion

Ref	Expression
bi2anan9r	⊢ ((θ ∧ φ) → ((ψ ∧ τ) ↔ (χ ∧ η)))

Proof of Theorem bi2anan9r

Step	Hyp	Ref	Expression
1		bi2an9.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2		bi2an9.2	. . 3 ⊢ (θ → (τ ↔ η))
3	1, 2	bi2anan9 478	. 2 ⊢ ((φ ∧ θ) → ((ψ ∧ τ) ↔ (χ ∧ η)))
4	3	ancoms 334	1 ⊢ ((θ ∧ φ) → ((ψ ∧ τ) ↔ (χ ∧ η)))

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

This theorem is referenced by:  axinfnd 3752  ltsopq 3869  ltsosr 3997  lt2sq 4414

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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