Theorem pm5.21ndd 621

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 40	. . 3 ⊢ (𝜑 → (𝜒 → (𝜒 ↔ 𝜃)))
4	3	ibd 167	. 2 ⊢ (𝜑 → (𝜒 → 𝜃))
5		pm5.21ndd.2	. . . . 5 ⊢ (𝜑 → (𝜃 → 𝜓))
6	5, 2	syld 40	. . . 4 ⊢ (𝜑 → (𝜃 → (𝜒 ↔ 𝜃)))
7		bicom1 122	. . . 4 ⊢ ((𝜒 ↔ 𝜃) → (𝜃 ↔ 𝜒))
8	6, 7	syl6 29	. . 3 ⊢ (𝜑 → (𝜃 → (𝜃 ↔ 𝜒)))
9	8	ibd 167	. 2 ⊢ (𝜑 → (𝜃 → 𝜒))
10	4, 9	impbid 120	1 ⊢ (𝜑 → (𝜒 ↔ 𝜃))

Syntax hints:   → wi 4   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  pm5.21nd  825  sbcrext  2835  rmob  2850  epelg  4027  eqbrrdva  4505  relbrcnvg  4704  fmptco  5330  ovelrn  5649  brtpos2  5866  brdomg  6229  genpelvl  6610  genpelvu  6611  fzoval  9005  clim  9802