Theorem biadan2 460

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 396	. 2 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜑))
3		biadan2.2	. . 3 ⊢ (𝜓 → (𝜑 ↔ 𝜒))
4	3	pm5.32i 458	. 2 ⊢ ((𝜓 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒))
5	2, 4	bitri 184	1 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  elab4g  2975  elpwb  3699  ssdifsn  3842  brab2a  4828  brab2ga  4850  elovmpo  6288  eqop2  6412  elnnnn0  9606  elixx3g  10303  elfzo2  10557  1nprm  12892