Theorem nanbi12 1297

Description: Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)

Assertion

Ref	Expression
nanbi12	⊢ (((φ ↔ ψ) ∧ (χ ↔ θ)) → ((φ ⊼ χ) ↔ (ψ ⊼ θ)))

Proof of Theorem nanbi12

Step	Hyp	Ref	Expression
1		nanbi1 1295	. 2 ⊢ ((φ ↔ ψ) → ((φ ⊼ χ) ↔ (ψ ⊼ χ)))
2		nanbi2 1296	. 2 ⊢ ((χ ↔ θ) → ((ψ ⊼ χ) ↔ (ψ ⊼ θ)))
3	1, 2	sylan9bb 680	1 ⊢ (((φ ↔ ψ) ∧ (χ ↔ θ)) → ((φ ⊼ χ) ↔ (ψ ⊼ θ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ⊼ wnan 1287

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 177  df-an 360  df-nan 1288

This theorem is referenced by:  nanbi12i  1300  nanbi12d  1303