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  5862  fvelimab  6981  brrpssg  7745  ordsucelsuc  7842  releldm2  8068  relbrtpos  8262  relelec  8792  elfiun  9470  fpwwe2lem2  10672  fpwwelem  10685  fzrev3  13630  elfzp12  13643  eqgval  19195  ismhp  22144  eltg  22964  eltg2  22965  cncnp2  23289  isref  23517  islocfin  23525  opeldifid  32612  isfne  36340  copsex2b  37141  bj-ideqgALT  37159  bj-idreseq  37163  bj-ideqg1ALT  37166  opelopab3  37725  isdivrngo  37957  brssr  38502  islshpkrN  39121  dihatexv2  41341