Theorem im2anan9r 435

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
im2anan9r	⊢ ((θ ∧ φ) → ((ψ ∧ τ) → (χ ∧ η)))

Proof of Theorem im2anan9r

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

Syntax hints:   → wi 2   ∧ wa 196

This theorem is referenced by:  wereu 2197  pssnn 3428

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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