Theorem a1bi 172

Description: Inference rule introducing a theorem as an antecedent.

Hypothesis

Ref	Expression
a1bi.1	⊢ φ

Assertion

Ref	Expression
a1bi	⊢ (ψ ↔ (φ → ψ))

Proof of Theorem a1bi

Step	Hyp	Ref	Expression
1		ax-1 3	. 2 ⊢ (ψ → (φ → ψ))
2		a1bi.1	. . 3 ⊢ φ
3		pm2.27 30	. . 3 ⊢ (φ → ((φ → ψ) → ψ))
4	2, 3	ax-mp 6	. 2 ⊢ ((φ → ψ) → ψ)
5	1, 4	impbi 139	1 ⊢ (ψ ↔ (φ → ψ))

Syntax hints:   → wi 2   ↔ wb 127

This theorem is referenced by:  sbequ8 902  ralv 1357  hbsbcv 1447  pw2en 3348

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

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