Theorem pm5.21nd 798

Description: Eliminate an antecedent implied by each side of a biconditional. Variant of pm5.21ndd 380. (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 412	. 2 ⊢ (𝜑 → (𝜓 → 𝜃))
3		pm5.21nd.2	. . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
4	3	ex 412	. 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 380	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395

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 206  df-an 396

This theorem is referenced by:  ideqg  5757  fvelimab  6835  brrpssg  7569  ordsucelsuc  7657  releldm2  7870  relbrtpos  8037  relelec  8517  elfiun  9150  fpwwe2lem2  10372  fpwwelem  10385  fzrev3  13304  elfzp12  13317  eqgval  18786  eltg  22088  eltg2  22089  cncnp2  22413  isref  22641  islocfin  22649  opeldifid  30917  isfne  34507  copsex2b  35290  bj-ideqgALT  35308  bj-idreseq  35312  bj-ideqg1ALT  35315  opelopab3  35854  isdivrngo  36087  brssr  36598  islshpkrN  37113  dihatexv2  39332