Theorem im2anan9 808

Description: Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)

Hypotheses

Ref	Expression
im2an9.1	⊢ (φ → (ψ → χ))
im2an9.2	⊢ (θ → (τ → η))

Assertion

Ref	Expression
im2anan9	⊢ ((φ ∧ θ) → ((ψ ∧ τ) → (χ ∧ η)))

Proof of Theorem im2anan9

Step	Hyp	Ref	Expression
1		im2an9.1	. . 3 ⊢ (φ → (ψ → χ))
2	1	adantr 451	. 2 ⊢ ((φ ∧ θ) → (ψ → χ))
3		im2an9.2	. . 3 ⊢ (θ → (τ → η))
4	3	adantl 452	. 2 ⊢ ((φ ∧ θ) → (τ → η))
5	2, 4	anim12d 546	1 ⊢ ((φ ∧ θ) → ((ψ ∧ τ) → (χ ∧ η)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360

This theorem is used by:  im2anan9r  809  ax11eq  2193  ax11el  2194  xpss12  4856  f1oun  5305  fntxp  5805  fnpprod  5844  fce  6189