Theorem biantrud 545

Description: A wff is equivalent to its conjunction with truth.

Hypothesis

Ref	Expression
biantrud.1	⊢ (φ → ψ)

Assertion

Ref	Expression
biantrud	⊢ (φ → (χ ↔ (χ ∧ ψ)))

Proof of Theorem biantrud

Step	Hyp	Ref	Expression
1		biantrud.1	. . . . 5 ⊢ (φ → ψ)
2	1	anim2i 270	. . . 4 ⊢ ((χ ∧ φ) → (χ ∧ ψ))
3	2	exp 291	. . 3 ⊢ (χ → (φ → (χ ∧ ψ)))
4	3	com12 13	. 2 ⊢ (φ → (χ → (χ ∧ ψ)))
5		pm3.26 256	. . 3 ⊢ ((χ ∧ ψ) → χ)
6	5	a1i 7	. 2 ⊢ (φ → ((χ ∧ ψ) → χ))
7	4, 6	impbid 397	1 ⊢ (φ → (χ ↔ (χ ∧ ψ)))

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

This theorem is referenced by:  biantrurd 546  mapdom2 3389  cardval 3633  nn2get 4438  shle0t 5367

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