Theorem im2anan9 620

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 491	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜏) → 𝜒))
3		im2an9.2	. . 3 ⊢ (𝜃 → (𝜏 → 𝜂))
4	3	adantld 490	. 2 ⊢ (𝜃 → ((𝜓 ∧ 𝜏) → 𝜂))
5	2, 4	anim12ii 618	1 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜏) → (𝜒 ∧ 𝜂)))

Syntax hints:   → wi 4   ∧ wa 395

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 207  df-an 396

This theorem is referenced by:  im2anan9r  621  anim12  808  mo4  2559  trin  5213  somo  5570  xpss12  5638  f1oun  6787  poxp  8068  soxp  8069  brecop  8744  dfac5lem4  10039  ingru  10728  genpss  10917  genpnnp  10918  tgcl  22873  txlm  23552  upgrpredgv  29103  3wlkdlem4  30125  frgrwopreglem5  30284  frgrwopreglem5ALT  30285  icorempo  37344  ax12eq  38939  ax12el  38940  odd2prm2  47722