Theorem ibib 448

Description: Implication in terms of implication and biconditional.

Assertion

Ref	Expression
ibib	⊢ ((φ → ψ) ↔ (φ → (φ ↔ ψ)))

Proof of Theorem ibib

Step	Hyp	Ref	Expression
1		pm3.4 266	. . . . 5 ⊢ ((φ ∧ ψ) → (φ → ψ))
2		pm3.26 256	. . . . . 6 ⊢ ((φ ∧ ψ) → φ)
3	2	a1d 14	. . . . 5 ⊢ ((φ ∧ ψ) → (ψ → φ))
4	1, 3	impbid 397	. . . 4 ⊢ ((φ ∧ ψ) → (φ ↔ ψ))
5	4	exp 291	. . 3 ⊢ (φ → (ψ → (φ ↔ ψ)))
6		bi1 130	. . . 4 ⊢ ((φ ↔ ψ) → (φ → ψ))
7	6	com12 13	. . 3 ⊢ (φ → ((φ ↔ ψ) → ψ))
8	5, 7	impbid 397	. 2 ⊢ (φ → (ψ ↔ (φ ↔ ψ)))
9	8	pm5.74i 443	1 ⊢ ((φ → ψ) ↔ (φ → (φ ↔ ψ)))

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

This theorem is referenced by:  ibd 451  oibabs 493  pm5.1 501  reuuni4 1959  brinxp 2466  zneo 4601

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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