Theorem pm5.21nd 802

Description: Eliminate an antecedent implied by each side of a biconditional. Variant of pm5.21ndd 379. (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 379	1 ⊢ (𝜑 → (𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ 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 207  df-an 396

This theorem is referenced by:  ideqg  5864  fvelimab  6980  brrpssg  7743  ordsucelsuc  7841  releldm2  8066  relbrtpos  8260  relelec  8790  elfiun  9467  fpwwe2lem2  10669  fpwwelem  10682  fzrev3  13626  elfzp12  13639  eqgval  19207  ismhp  22161  eltg  22979  eltg2  22980  cncnp2  23304  isref  23532  islocfin  23540  opeldifid  32618  isfne  36321  copsex2b  37122  bj-ideqgALT  37140  bj-idreseq  37144  bj-ideqg1ALT  37147  opelopab3  37704  isdivrngo  37936  brssr  38482  islshpkrN  39101  dihatexv2  41321