Theorem pm5.21ndd 712

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  923  sbcrext  3109  rmob  3125  epelg  4387  eqbrrdva  4900  elrelimasn  5102  relbrcnvg  5115  fmptco  5813  ovelrn  6171  brtpos2  6417  elpmg  6833  brdomg  6919  elfi2  7171  genpelvl  7732  genpelvu  7733  fzoval  10383  nninfinf  10706  clim  11843  dvdsaddre2b  12404  pceu  12870  divsfval  13413  sgrppropd  13498  mndpropd  13525  issubg3  13781  resghm2b  13851  rngpropd  13971  dvdsrd  14111  opprsubrngg  14228  subrngpropd  14233  subrgpropd  14270  rhmpropd  14271  lmodprop2d  14365  cnrest2  14963  cnptoprest2  14967  lmss  14973  reopnap  15273  limcdifap  15389  iswlkg  16183  isclwwlkng  16260