Theorem anim12dan 810

Description: Conjoin antecedents and consequents in a deduction. (Contributed by Mario Carneiro, 12-May-2014.)

Hypotheses

Ref	Expression
anim12dan.1	⊢ ((φ ∧ ψ) → χ)
anim12dan.2	⊢ ((φ ∧ θ) → τ)

Assertion

Ref	Expression
anim12dan	⊢ ((φ ∧ (ψ ∧ θ)) → (χ ∧ τ))

Proof of Theorem anim12dan

Step	Hyp	Ref	Expression
1		anim12dan.1	. . . 4 ⊢ ((φ ∧ ψ) → χ)
2	1	ex 423	. . 3 ⊢ (φ → (ψ → χ))
3		anim12dan.2	. . . 4 ⊢ ((φ ∧ θ) → τ)
4	3	ex 423	. . 3 ⊢ (φ → (θ → τ))
5	2, 4	anim12d 546	. 2 ⊢ (φ → ((ψ ∧ θ) → (χ ∧ τ)))
6	5	imp 418	1 ⊢ ((φ ∧ (ψ ∧ θ)) → (χ ∧ τ))

Syntax hints:   → wi 4   ∧ wa 358

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 177  df-an 360

This theorem is referenced by:  xpexr2  5110  isocnv  5491  f1oiso2  5500