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  721  prlem1  975  equveli  1781  elssabg  4191  suctr  4467  fvun1  5644  opabbrex  5988  poxp  6317  tposfo2  6352  1idprl  7702  1idpru  7703  uzind  9483  xrlttr  9916  fzen  10164  fz0fzelfz0  10248  ccatsymb  11056  fisumss  11645  fprodssdc  11843  zeqzmulgcd  12233  lcmgcdlem  12341  lcmdvds  12343  cncongr2  12368  exprmfct  12402  pceu  12560  infpnlem1  12624  isghm  13521  ringadd2  13731  metrest  14920  bj-charfunbi  15680  bj-om  15806