Theorem im3an 605

Description: Join antecedents and consequents with conjunction.

Hypotheses

Ref	Expression
im3an.1	⊢ (φ → ψ)
im3an.2	⊢ (χ → θ)
im3an.3	⊢ (τ → η)

Assertion

Ref	Expression
im3an	⊢ ((φ ∧ χ ∧ τ) → (ψ ∧ θ ∧ η))

Proof of Theorem im3an

Step	Hyp	Ref	Expression
1		im3an.1	. . . 4 ⊢ (φ → ψ)
2		im3an.2	. . . 4 ⊢ (χ → θ)
3	1, 2	anim12i 268	. . 3 ⊢ ((φ ∧ χ) → (ψ ∧ θ))
4		im3an.3	. . 3 ⊢ (τ → η)
5	3, 4	anim12i 268	. 2 ⊢ (((φ ∧ χ) ∧ τ) → ((ψ ∧ θ) ∧ η))
6		df-3an 583	. 2 ⊢ ((φ ∧ χ ∧ τ) ↔ ((φ ∧ χ) ∧ τ))
7		df-3an 583	. 2 ⊢ ((ψ ∧ θ ∧ η) ↔ ((ψ ∧ θ) ∧ η))
8	5, 6, 7	3imtr4 192	1 ⊢ ((φ ∧ χ ∧ τ) → (ψ ∧ θ ∧ η))

Syntax hints:   → wi 2   ∧ wa 196   ∧ w3a 581

This theorem is referenced by:  syl3an 628  eloprabg 3035  distrlem3pr 3923  divasst 4239  le2tri3 4311  nnleltp1t 4448  atcvatlem 5770

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583

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