Theorem pm5.21ndd 709

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  920  sbcrext  3086  rmob  3102  epelg  4358  eqbrrdva  4869  elrelimasn  5070  relbrcnvg  5083  fmptco  5774  ovelrn  6125  brtpos2  6367  elpmg  6781  brdomg  6867  elfi2  7107  genpelvl  7667  genpelvu  7668  fzoval  10312  nninfinf  10632  clim  11758  dvdsaddre2b  12318  pceu  12784  divsfval  13327  sgrppropd  13412  mndpropd  13439  issubg3  13695  resghm2b  13765  rngpropd  13884  dvdsrd  14023  opprsubrngg  14140  subrngpropd  14145  subrgpropd  14182  rhmpropd  14183  lmodprop2d  14277  cnrest2  14875  cnptoprest2  14879  lmss  14885  reopnap  15185  limcdifap  15301