Theorem confun 46885

Description: Given the hypotheses there exists a proof for (c implies ( d iff a ) ). (Contributed by Jarvin Udandy, 6-Sep-2020.)

Hypotheses

Ref	Expression
confun.1	⊢ 𝜑
confun.2	⊢ (𝜒 → 𝜓)
confun.3	⊢ (𝜒 → 𝜃)
confun.4	⊢ (𝜑 → (𝜑 → 𝜓))

Assertion

Ref	Expression
confun	⊢ (𝜒 → (𝜃 ↔ 𝜑))

Proof of Theorem confun

Step	Hyp	Ref	Expression
1		ax-1 6	. . 3 ⊢ (𝜒 → (𝜃 → 𝜒))
2		confun.3	. . . 4 ⊢ (𝜒 → 𝜃)
3	2	a1i 11	. . 3 ⊢ (𝜒 → (𝜒 → 𝜃))
4	1, 3	impbid 212	. 2 ⊢ (𝜒 → (𝜃 ↔ 𝜒))
5		confun.2	. . . . 5 ⊢ (𝜒 → 𝜓)
6		confun.1	. . . . . . 7 ⊢ 𝜑
7		confun.4	. . . . . . 7 ⊢ (𝜑 → (𝜑 → 𝜓))
8	6, 7	ax-mp 5	. . . . . 6 ⊢ (𝜑 → 𝜓)
9		ax-1 6	. . . . . . 7 ⊢ (𝜑 → (𝜓 → 𝜑))
10	6, 9	ax-mp 5	. . . . . 6 ⊢ (𝜓 → 𝜑)
11	8, 10	impbii 209	. . . . 5 ⊢ (𝜑 ↔ 𝜓)
12	5, 11	sylibr 234	. . . 4 ⊢ (𝜒 → 𝜑)
13	12	a1i 11	. . 3 ⊢ (𝜒 → (𝜒 → 𝜑))
14		ax-1 6	. . 3 ⊢ (𝜒 → (𝜑 → 𝜒))
15	13, 14	impbid 212	. 2 ⊢ (𝜒 → (𝜒 ↔ 𝜑))
16	4, 15	bitrd 279	1 ⊢ (𝜒 → (𝜃 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207

This theorem is referenced by:  confun2  46886  confun3  46887