Theorem biadan2 445

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 385	. 2 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑))
3		biadan2.2	. . 3 ⊢ (𝜓 → (𝜑 ↔ 𝜒))
4	3	pm5.32i 443	. 2 ⊢ ((𝜓 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒))
5	2, 4	bitri 183	1 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  elab4g  2765  elpwb  3443  ssdifsn  3574  brab2a  4504  brab2ga  4526  elovmpt2  5859  eqop2  5962  elnnnn0  8777  elixx3g  9380  elfzo2  9622  1nprm  11435