Theorem 3ad2antl1 1185

Description: Deduction adding conjuncts to antecedent. (Contributed by NM, 4-Aug-2007.)

Hypothesis

Ref	Expression
3ad2antl.1	⊢ ((𝜑 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
3ad2antl1	⊢ (((𝜑 ∧ 𝜓 ∧ 𝜏) ∧ 𝜒) → 𝜃)

Proof of Theorem 3ad2antl1

Step	Hyp	Ref	Expression
1		3ad2antl.1	. . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
2	1	adantlr 477	. 2 ⊢ (((𝜑 ∧ 𝜏) ∧ 𝜒) → 𝜃)
3	2	3adantl2 1180	1 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜏) ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 1006

This theorem is referenced by:  acexmid  6017  f1oen4g  6925  f1dom4g  6926  ordiso2  7234  addlocpr  7756  distrlem1prl  7802  distrlem1pru  7803  ltsopr  7816  addcanprlemu  7835  fzo1fzo0n0  10423  pfxsuffeqwrdeq  11283  prodfap0  12111  prodfrecap  12112  muldvds2  12383  dvds2add  12391  dvds2sub  12392  dvdstr  12394  qusaddvallemg  13421  mulgnnsubcl  13726  mulgpropdg  13756  ringidss  14048  lmodprop2d  14368  cnpnei  14949  upxp  15002  lgsval4lem  15746  clwwlkccatlem  16257  clwwlkccat  16258