Theorem anim12d1 610

Description: Variant of anim12d 609 where the second implication does not depend on the antecedent. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Hypotheses

Ref	Expression
anim12d1.1	⊢ (𝜑 → (𝜓 → 𝜒))
anim12d1.2	⊢ (𝜃 → 𝜏)

Assertion

Ref	Expression
anim12d1	⊢ (𝜑 → ((𝜓 ∧ 𝜃) → (𝜒 ∧ 𝜏)))

Proof of Theorem anim12d1

Step	Hyp	Ref	Expression
1		anim12d1.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
2		anim12d1.2	. . 3 ⊢ (𝜃 → 𝜏)
3	2	a1i 11	. 2 ⊢ (𝜑 → (𝜃 → 𝜏))
4	1, 3	anim12d 609	1 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → (𝜒 ∧ 𝜏)))

Syntax hints:   → wi 4   ∧ wa 396

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 206  df-an 397

This theorem is referenced by:  fun  6636  frrlem13  8114  alephord  9831  grudomon  10573  xrsupexmnf  13039  xrinfmexpnf  13040  joinfval  18091  meetfval  18105  cnpresti  22439  1stcrest  22604  upgrwlkdvdelem  28104