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Theorem pldofph 40446
Description: Given, a,b c, d, "definition" for e, e is demonstrated. (Contributed by Jarvin Udandy, 8-Sep-2020.)
Hypotheses
Ref Expression
pldofph.1 (𝜏 ↔ ((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃))))
pldofph.2 𝜑
pldofph.3 𝜓
pldofph.4 𝜒
pldofph.5 𝜃
Assertion
Ref Expression
pldofph 𝜏

Proof of Theorem pldofph
StepHypRef Expression
1 pldofph.5 . . . 4 𝜃
21a1i 11 . . 3 (𝜒𝜃)
3 pldofph.2 . . . 4 𝜑
4 pldofph.4 . . . 4 𝜒
53, 42th 254 . . 3 (𝜑𝜒)
6 pldofph.3 . . . . 5 𝜓
76, 12th 254 . . . 4 (𝜓𝜃)
87a1i 11 . . 3 ((𝜑𝜓) → (𝜓𝜃))
92, 5, 83pm3.2i 1237 . 2 ((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃)))
10 pldofph.1 . . . 4 (𝜏 ↔ ((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃))))
1110bicomi 214 . . 3 (((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃))) ↔ 𝜏)
1211biimpi 206 . 2 (((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃))) → 𝜏)
139, 12ax-mp 5 1 𝜏
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036
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  df-an 386  df-3an 1038
This theorem is referenced by:  plvcofph  40447  plvcofphax  40448
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