Theorem biadan2 623

Description: Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.)

Hypotheses

Ref	Expression
biadan2.1	⊢ (φ → ψ)
biadan2.2	⊢ (ψ → (φ ↔ χ))

Assertion

Ref	Expression
biadan2	⊢ (φ ↔ (ψ ∧ χ))

Proof of Theorem biadan2

Step	Hyp	Ref	Expression
1		biadan2.1	. . 3 ⊢ (φ → ψ)
2	1	pm4.71ri 614	. 2 ⊢ (φ ↔ (ψ ∧ φ))
3		biadan2.2	. . 3 ⊢ (ψ → (φ ↔ χ))
4	3	pm5.32i 618	. 2 ⊢ ((ψ ∧ φ) ↔ (ψ ∧ χ))
5	2, 4	bitri 240	1 ⊢ (φ ↔ (ψ ∧ χ))

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

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

This theorem is referenced by:  elab4g  2989  eqpw1  4162  eqnc2  6129