Theorem elimh 1081

Description: Hypothesis builder for 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
elimh.1	⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜏 ↔ 𝜒))
elimh.2	⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜓) → (𝜏 ↔ 𝜃))
elimh.3	⊢ 𝜃

Assertion

Ref	Expression
elimh	⊢ 𝜏

Proof of Theorem elimh

Step	Hyp	Ref	Expression
1		ifptru 1072	. . . 4 ⊢ (𝜒 → (if-(𝜒, 𝜑, 𝜓) ↔ 𝜑))
2		elimh.1	. . . 4 ⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜏 ↔ 𝜒))
3	1, 2	syl 17	. . 3 ⊢ (𝜒 → (𝜏 ↔ 𝜒))
4	3	ibir 267	. 2 ⊢ (𝜒 → 𝜏)
5		elimh.3	. . 3 ⊢ 𝜃
6		ifpfal 1073	. . . 4 ⊢ (¬ 𝜒 → (if-(𝜒, 𝜑, 𝜓) ↔ 𝜓))
7		elimh.2	. . . 4 ⊢ ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜓) → (𝜏 ↔ 𝜃))
8	6, 7	syl 17	. . 3 ⊢ (¬ 𝜒 → (𝜏 ↔ 𝜃))
9	5, 8	mpbiri 257	. 2 ⊢ (¬ 𝜒 → 𝜏)
10	4, 9	pm2.61i 182	1 ⊢ 𝜏

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  if-wif 1059

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 206  df-an 396  df-or 844  df-ifp 1060

This theorem is referenced by:  con3ALT  1083