Theorem pm3.2 232

Description: Join antecedents with conjunction. Theorem *3.2 of [WhiteheadRussell] p. 111.

Assertion

Ref	Expression
pm3.2	⊢ (φ → (ψ → (φ ∧ ψ)))

Proof of Theorem pm3.2

Step	Hyp	Ref	Expression
1		df-an 198	. . 3 ⊢ ((φ ∧ ψ) ↔ ¬ (φ → ¬ ψ))
2	1	biimpr 134	. 2 ⊢ (¬ (φ → ¬ ψ) → (φ ∧ ψ))
3	2	expi 125	1 ⊢ (φ → (ψ → (φ ∧ ψ)))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196

This theorem is referenced by:  pm3.21 233  pm3.2i 234  pm3.43i 235  ancl 242  anc2l 248  anidm 331  prth 429  19.26 749  difrab 1695  indpi 3828  alephexp2 4956

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

metamath.org