Theorem im2anan9 434

Description: Deduction joining nested implications to form implication of conjunctions.

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 306	. 2 ⊢ ((φ ∧ θ) → (ψ → χ))
3		im2an9.2	. . 3 ⊢ (θ → (τ → η))
4	3	adantl 305	. 2 ⊢ ((φ ∧ θ) → (τ → η))
5	2, 4	anim12d 431	1 ⊢ ((φ ∧ θ) → ((ψ ∧ τ) → (χ ∧ η)))

Syntax hints:   → wi 2   ∧ wa 196

This theorem is referenced by:  trin 2051  f1oun 2815  brecop 3242  genpss 3901  genpnnp 3902  distrlem1pr 3921  ssgt0sr 4011  uzwo2 4606  shselt 5280

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

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