Theorem pm5.21ndd 710

Description: Eliminate an antecedent implied by each side of a biconditional, deduction version. (Contributed by Paul Chapman, 21-Nov-2012.) (Revised by Mario Carneiro, 31-Jan-2015.)

Hypotheses

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

Assertion

Ref	Expression
pm5.21ndd	⊢ (𝜑 → (𝜒 ↔ 𝜃))

Proof of Theorem pm5.21ndd

Step	Hyp	Ref	Expression
1		pm5.21ndd.1	. . . 4 ⊢ (𝜑 → (𝜒 → 𝜓))
2		pm5.21ndd.3	. . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))
3	1, 2	syld 45	. . 3 ⊢ (𝜑 → (𝜒 → (𝜒 ↔ 𝜃)))
4	3	ibd 178	. 2 ⊢ (𝜑 → (𝜒 → 𝜃))
5		pm5.21ndd.2	. . . . 5 ⊢ (𝜑 → (𝜃 → 𝜓))
6	5, 2	syld 45	. . . 4 ⊢ (𝜑 → (𝜃 → (𝜒 ↔ 𝜃)))
7		bicom1 131	. . . 4 ⊢ ((𝜒 ↔ 𝜃) → (𝜃 ↔ 𝜒))
8	6, 7	syl6 33	. . 3 ⊢ (𝜑 → (𝜃 → (𝜃 ↔ 𝜒)))
9	8	ibd 178	. 2 ⊢ (𝜑 → (𝜃 → 𝜒))
10	4, 9	impbid 129	1 ⊢ (𝜑 → (𝜒 ↔ 𝜃))

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm5.21nd  921  sbcrext  3106  rmob  3122  epelg  4382  eqbrrdva  4895  elrelimasn  5097  relbrcnvg  5110  fmptco  5806  ovelrn  6163  brtpos2  6408  elpmg  6824  brdomg  6910  elfi2  7155  genpelvl  7715  genpelvu  7716  fzoval  10361  nninfinf  10682  clim  11813  dvdsaddre2b  12373  pceu  12839  divsfval  13382  sgrppropd  13467  mndpropd  13494  issubg3  13750  resghm2b  13820  rngpropd  13939  dvdsrd  14079  opprsubrngg  14196  subrngpropd  14201  subrgpropd  14238  rhmpropd  14239  lmodprop2d  14333  cnrest2  14931  cnptoprest2  14935  lmss  14941  reopnap  15241  limcdifap  15357  iswlkg  16101  isclwwlkng  16175