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Theorem bnj887 31215
Description: -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj887.1 (𝜑𝜑′)
bnj887.2 (𝜓𝜓′)
bnj887.3 (𝜒𝜒′)
bnj887.4 (𝜃𝜃′)
Assertion
Ref Expression
bnj887 ((𝜑𝜓𝜒𝜃) ↔ (𝜑′𝜓′𝜒′𝜃′))

Proof of Theorem bnj887
StepHypRef Expression
1 bnj887.1 . . . 4 (𝜑𝜑′)
2 bnj887.2 . . . 4 (𝜓𝜓′)
3 bnj887.3 . . . 4 (𝜒𝜒′)
41, 2, 33anbi123i 1194 . . 3 ((𝜑𝜓𝜒) ↔ (𝜑′𝜓′𝜒′))
5 bnj887.4 . . 3 (𝜃𝜃′)
64, 5anbi12i 620 . 2 (((𝜑𝜓𝜒) ∧ 𝜃) ↔ ((𝜑′𝜓′𝜒′) ∧ 𝜃′))
7 df-bnj17 31136 . 2 ((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓𝜒) ∧ 𝜃))
8 df-bnj17 31136 . 2 ((𝜑′𝜓′𝜒′𝜃′) ↔ ((𝜑′𝜓′𝜒′) ∧ 𝜃′))
96, 7, 83bitr4i 294 1 ((𝜑𝜓𝜒𝜃) ↔ (𝜑′𝜓′𝜒′𝜃′))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  w3a 1107  w-bnj17 31135
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 198  df-an 385  df-3an 1109  df-bnj17 31136
This theorem is referenced by:  bnj1040  31420  bnj1128  31438
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