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Theorem confun 40869
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 202 . 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 199 . . . . 5 (𝜑𝜓)
125, 11sylibr 224 . . . 4 (𝜒𝜑)
1312a1i 11 . . 3 (𝜒 → (𝜒𝜑))
14 ax-1 6 . . 3 (𝜒 → (𝜑𝜒))
1513, 14impbid 202 . 2 (𝜒 → (𝜒𝜑))
164, 15bitrd 268 1 (𝜒 → (𝜃𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196
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 197
This theorem is referenced by:  confun2  40870  confun3  40871
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