Theorem adantrd 279

Description: Deduction adding a conjunct to the right of an antecedent. (Contributed by NM, 4-May-1994.)

Hypothesis

Ref	Expression
adantrd.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
adantrd	⊢ (𝜑 → ((𝜓 ∧ 𝜃) → 𝜒))

Proof of Theorem adantrd

Step	Hyp	Ref	Expression
1		simpl 109	. 2 ⊢ ((𝜓 ∧ 𝜃) → 𝜓)
2		adantrd.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3	1, 2	syl5 32	1 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → 𝜒))

Syntax hints:   → wi 4   ∧ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106

This theorem is referenced by:  syldan  282  jaoa  720  prlem1  973  equveli  1759  elssabg  4150  suctr  4423  fvun1  5584  opabbrex  5921  poxp  6235  tposfo2  6270  1idprl  7591  1idpru  7592  uzind  9366  xrlttr  9797  fzen  10045  fz0fzelfz0  10129  fisumss  11402  fprodssdc  11600  zeqzmulgcd  11973  lcmgcdlem  12079  lcmdvds  12081  cncongr2  12106  exprmfct  12140  pceu  12297  infpnlem1  12359  ringadd2  13215  metrest  14091  bj-charfunbi  14648  bj-om  14774