Theorem pm5.21ndd 653

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 44	. . 3 ⊢ (𝜑 → (𝜒 → (𝜒 ↔ 𝜃)))
4	3	ibd 176	. 2 ⊢ (𝜑 → (𝜒 → 𝜃))
5		pm5.21ndd.2	. . . . 5 ⊢ (𝜑 → (𝜃 → 𝜓))
6	5, 2	syld 44	. . . 4 ⊢ (𝜑 → (𝜃 → (𝜒 ↔ 𝜃)))
7		bicom1 129	. . . 4 ⊢ ((𝜒 ↔ 𝜃) → (𝜃 ↔ 𝜒))
8	6, 7	syl6 33	. . 3 ⊢ (𝜑 → (𝜃 → (𝜃 ↔ 𝜒)))
9	8	ibd 176	. 2 ⊢ (𝜑 → (𝜃 → 𝜒))
10	4, 9	impbid 127	1 ⊢ (𝜑 → (𝜒 ↔ 𝜃))

Syntax hints:   → wi 4   ↔ wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  pm5.21nd  858  sbcrext  2891  rmob  2906  epelg  4045  eqbrrdva  4523  relbrcnvg  4724  fmptco  5351  ovelrn  5669  brtpos2  5889  brdomg  6252  genpelvl  6702  genpelvu  6703  fzoval  9158  clim  10120