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Theorem confun 39755
 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 (𝜒𝜓)
confun.3 (𝜒𝜃)
confun.4 (𝜑 → (𝜑𝜓))
Assertion
Ref Expression
confun (𝜒 → (𝜃𝜑))

Proof of Theorem confun
StepHypRef Expression
1 ax-1 6 . . 3 (𝜒 → (𝜃𝜒))
2 confun.3 . . . 4 (𝜒𝜃)
32a1i 11 . . 3 (𝜒 → (𝜒𝜃))
41, 3impbid 201 . 2 (𝜒 → (𝜃𝜒))
5 confun.2 . . . . 5 (𝜒𝜓)
6 confun.1 . . . . . . 7 𝜑
7 confun.4 . . . . . . 7 (𝜑 → (𝜑𝜓))
86, 7ax-mp 5 . . . . . 6 (𝜑𝜓)
9 ax-1 6 . . . . . . 7 (𝜑 → (𝜓𝜑))
106, 9ax-mp 5 . . . . . 6 (𝜓𝜑)
118, 10impbii 198 . . . . 5 (𝜑𝜓)
125, 11sylibr 223 . . . 4 (𝜒𝜑)
1312a1i 11 . . 3 (𝜒 → (𝜒𝜑))
14 ax-1 6 . . 3 (𝜒 → (𝜑𝜒))
1513, 14impbid 201 . 2 (𝜒 → (𝜒𝜑))
164, 15bitrd 267 1 (𝜒 → (𝜃𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195 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 196 This theorem is referenced by:  confun2  39756  confun3  39757
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