Theorem dedt 1084

Description: The weak deduction theorem. For more information, see the Weak Deduction Theorem page mmdeduction.html. (Contributed by NM, 26-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) Commute consequent. (Revised by Steven Nguyen, 27-Apr-2023.)

Hypotheses

Ref	Expression
dedt.1	⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜏 ↔ 𝜃))
dedt.2	⊢ 𝜏

Assertion

Ref	Expression
dedt	⊢ (𝜒 → 𝜃)

Proof of Theorem dedt

Step	Hyp	Ref	Expression
1		ifptru 1075	. 2 ⊢ (𝜒 → (if-(𝜒, 𝜑, 𝜓) ↔ 𝜑))
2		dedt.2	. . 3 ⊢ 𝜏
3		dedt.1	. . 3 ⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜏 ↔ 𝜃))
4	2, 3	mpbii 233	. 2 ⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → 𝜃)
5	1, 4	syl 17	1 ⊢ (𝜒 → 𝜃)

Syntax hints:   → wi 4   ↔ wb 206  if-wif 1063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064

This theorem is referenced by:  con3ALT  1085