Theorem pm5.21nd 798

Description: Eliminate an antecedent implied by each side of a biconditional. Variant of pm5.21ndd 381. (Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen, 4-Nov-2013.)

Hypotheses

Ref	Expression
pm5.21nd.1	⊢ ((𝜑 ∧ 𝜓) → 𝜃)
pm5.21nd.2	⊢ ((𝜑 ∧ 𝜒) → 𝜃)
pm5.21nd.3	⊢ (𝜃 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
pm5.21nd	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Proof of Theorem pm5.21nd

Step	Hyp	Ref	Expression
1		pm5.21nd.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
2	1	ex 413	. 2 ⊢ (𝜑 → (𝜓 → 𝜃))
3		pm5.21nd.2	. . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
4	3	ex 413	. 2 ⊢ (𝜑 → (𝜒 → 𝜃))
5		pm5.21nd.3	. . 3 ⊢ (𝜃 → (𝜓 ↔ 𝜒))
6	5	a1i 11	. 2 ⊢ (𝜑 → (𝜃 → (𝜓 ↔ 𝜒)))
7	2, 4, 6	pm5.21ndd 381	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396

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 208  df-an 397

This theorem is referenced by:  ideqg  5600  fvelimab  6597  brrpssg  7300  ordsucelsuc  7384  releldm2  7589  relbrtpos  7745  relelec  8175  elfiun  8730  fpwwe2lem2  9889  fpwwelem  9902  fzrev3  12812  elfzp12  12825  eqgval  18070  eltg  21237  eltg2  21238  cncnp2  21561  isref  21789  islocfin  21797  opeldifid  30015  isfne  33241  opelopab3  34470  isdivrngo  34706  brssr  35222  islshpkrN  35737  dihatexv2  37956