Theorem abai 366

Description: Introduce one conjunct as an antecedent to the another.

Assertion

Ref	Expression
abai	⊢ ((φ ∧ ψ) ↔ (φ ∧ (φ → ψ)))

Proof of Theorem abai

Step	Hyp	Ref	Expression
1		pm3.26 256	. . 3 ⊢ ((φ ∧ ψ) → φ)
2		pm3.4 266	. . 3 ⊢ ((φ ∧ ψ) → (φ → ψ))
3	1, 2	jca 236	. 2 ⊢ ((φ ∧ ψ) → (φ ∧ (φ → ψ)))
4		pm3.26 256	. . 3 ⊢ ((φ ∧ (φ → ψ)) → φ)
5		pm3.35 278	. . 3 ⊢ ((φ ∧ (φ → ψ)) → ψ)
6	4, 5	jca 236	. 2 ⊢ ((φ ∧ (φ → ψ)) → (φ ∧ ψ))
7	3, 6	impbi 139	1 ⊢ ((φ ∧ ψ) ↔ (φ ∧ (φ → ψ)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196

This theorem is referenced by:  eu2 1023  euanv 1053  r19.29 1295  dfss4 1667  difin 1670  tfrlem2 2950  choc0 5291

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

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