Theorem adantrd 273

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 107	. 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 102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104

This theorem is referenced by:  syldan  276  jaoa  673  prlem1  917  equveli  1686  elssabg  3958  suctr  4221  fvun1  5327  opabbrex  5644  poxp  5948  tposfo2  5980  1idprl  7086  1idpru  7087  uzind  8783  xrlttr  9190  fzen  9382  fz0fzelfz0  9459  zeqzmulgcd  10829  lcmgcdlem  10926  lcmdvds  10928  cncongr2  10953  exprmfct  10986  bj-om  11262