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Theorem 3anim123d 1213
Description: Deduction joining 3 implications to form implication of conjunctions. (Contributed by NM, 24-Feb-2005.)
Hypotheses
Ref Expression
3anim123d.1 (φ → (ψχ))
3anim123d.2 (φ → (θτ))
3anim123d.3 (φ → (ηζ))
Assertion
Ref Expression
3anim123d (φ → ((ψ θ η) → (χ τ ζ)))

Proof of Theorem 3anim123d
StepHypRef Expression
1 3anim123d.1 . . . 4 (φ → (ψχ))
2 3anim123d.2 . . . 4 (φ → (θτ))
31, 2anim12d 318 . . 3 (φ → ((ψ θ) → (χ τ)))
4 3anim123d.3 . . 3 (φ → (ηζ))
53, 4anim12d 318 . 2 (φ → (((ψ θ) η) → ((χ τ) ζ)))
6 df-3an 886 . 2 ((ψ θ η) ↔ ((ψ θ) η))
7 df-3an 886 . 2 ((χ τ ζ) ↔ ((χ τ) ζ))
85, 6, 73imtr4g 194 1 (φ → ((ψ θ η) → (χ τ ζ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem depends on definitions:  df-bi 110  df-3an 886
This theorem is referenced by:  hb3and  1376  pofun  4040  soss  4042  isopolem  5404  isosolem  5406  issmo2  5845  smores  5848
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