Theorem biimt 549

Description: A wff is equivalent to itself with true antecedent.

Assertion

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

Proof of Theorem biimt

Step	Hyp	Ref	Expression
1		ax-1 3	. . 3 ⊢ (ψ → (φ → ψ))
2	1	a1i 7	. 2 ⊢ (φ → (ψ → (φ → ψ)))
3		pm2.27 30	. 2 ⊢ (φ → ((φ → ψ) → ψ))
4	2, 3	impbid 397	1 ⊢ (φ → (ψ ↔ (φ → ψ)))

Syntax hints:   → wi 2   ↔ wb 127

This theorem is referenced by:  biorf 551  sbc5g 1450  sbc6g 1451  elrabsf 1456  r19.3rzv 1767  ralidm 1774  brecop 3242  kmlem12 3591  kmlem13 3592

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