Theorem im2anan9 626

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	adantrd 492	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜏) → 𝜒))
3		im2an9.2	. . 3 ⊢ (𝜃 → (𝜏 → 𝜂))
4	3	adantld 491	. 2 ⊢ (𝜃 → ((𝜓 ∧ 𝜏) → 𝜂))
5	2, 4	anim12ii 624	1 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜏) → (𝜒 ∧ 𝜂)))

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 208  df-an 397

This theorem is referenced by:  im2anan9r  627  anim12  814  mo4  2570  trin  5191  somo  5565  xpss12  5633  f1oun  6786  poxp  8068  soxp  8069  brecop  8747  dfac5lem4  10039  ingru  10729  genpss  10918  genpnnp  10919  tgcl  22952  txlm  23631  upgrpredgv  29226  3wlkdlem4  30250  frgrwopreglem5  30409  frgrwopreglem5ALT  30410  icorempo  37713  ax12eq  39433  ax12el  39434  odd2prm2  48209