Theorem 3pm3.2i 1170

Description: Infer conjunction of premises. (Contributed by NM, 10-Feb-1995.)

Hypotheses

Ref	Expression
3pm3.2i.1	⊢ 𝜑
3pm3.2i.2	⊢ 𝜓
3pm3.2i.3	⊢ 𝜒

Assertion

Ref	Expression
3pm3.2i	⊢ (𝜑 ∧ 𝜓 ∧ 𝜒)

Proof of Theorem 3pm3.2i

Step	Hyp	Ref	Expression
1		3pm3.2i.1	. . 3 ⊢ 𝜑
2		3pm3.2i.2	. . 3 ⊢ 𝜓
3	1, 2	pm3.2i 270	. 2 ⊢ (𝜑 ∧ 𝜓)
4		3pm3.2i.3	. 2 ⊢ 𝜒
5		df-3an 975	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
6	3, 4, 5	mpbir2an 937	1 ⊢ (𝜑 ∧ 𝜓 ∧ 𝜒)

Syntax hints:   ∧ wa 103   ∧ w3a 973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 975

This theorem is referenced by:  mpbir3an  1174  3jaoi  1298  ftp  5681  4bc2eq6  10708  halfleoddlt  11853  strleun  12507  strle1g  12508  2irrexpqap  13690  lgslem2  13696  lgsdir2lem2  13724  lgsdir2lem3  13725  ex-dvds  13765  nconstwlpolem0  14094