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Theorem pldofph 39761
 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 253 . . 3 (𝜑𝜒)
6 pldofph.3 . . . . 5 𝜓
76, 12th 253 . . . 4 (𝜓𝜃)
87a1i 11 . . 3 ((𝜑𝜓) → (𝜓𝜃))
92, 5, 83pm3.2i 1232 . 2 ((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃)))
10 pldofph.1 . . . 4 (𝜏 ↔ ((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃))))
1110bicomi 213 . . 3 (((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃))) ↔ 𝜏)
1211biimpi 205 . 2 (((𝜒𝜃) ∧ (𝜑𝜒) ∧ ((𝜑𝜓) → (𝜓𝜃))) → 𝜏)
139, 12ax-mp 5 1 𝜏
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031 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  df-an 385  df-3an 1033 This theorem is referenced by:  plvcofph  39762  plvcofphax  39763
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