Theorem im2anan9 622

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 495	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜏) → 𝜒))
3		im2an9.2	. . 3 ⊢ (𝜃 → (𝜏 → 𝜂))
4	3	adantld 494	. 2 ⊢ (𝜃 → ((𝜓 ∧ 𝜏) → 𝜂))
5	2, 4	anim12ii 620	1 ⊢ ((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜏) → (𝜒 ∧ 𝜂)))

Syntax hints:   → wi 4   ∧ wa 399

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 210  df-an 400

This theorem is referenced by:  im2anan9r  623  anim12  808  trin  5148  somo  5479  xpss12  5539  f1oun  6621  poxp  7827  soxp  7828  brecop  8400  ingru  10275  genpss  10464  genpnnp  10465  tgcl  21669  txlm  22348  upgrpredgv  27031  3wlkdlem4  28046  frgrwopreglem5  28205  frgrwopreglem5ALT  28206  icorempo  35048  ax12eq  36517  ax12el  36518  odd2prm2  44603