Theorem nanbi12d 1303

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

Hypotheses

Ref	Expression
nanbid.1	⊢ (φ → (ψ ↔ χ))
nanbi12d.2	⊢ (φ → (θ ↔ τ))

Assertion

Ref	Expression
nanbi12d	⊢ (φ → ((ψ ⊼ θ) ↔ (χ ⊼ τ)))

Proof of Theorem nanbi12d

Step	Hyp	Ref	Expression
1		nanbid.1	. 2 ⊢ (φ → (ψ ↔ χ))
2		nanbi12d.2	. 2 ⊢ (φ → (θ ↔ τ))
3		nanbi12 1297	. 2 ⊢ (((ψ ↔ χ) ∧ (θ ↔ τ)) → ((ψ ⊼ θ) ↔ (χ ⊼ τ)))
4	1, 2, 3	syl2anc 642	1 ⊢ (φ → ((ψ ⊼ θ) ↔ (χ ⊼ τ)))

Syntax hints:   → wi 4   ↔ wb 176   ⊼ 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:  ninjust  3211  elning  3218  elun  3221