Theorem dedth2v 4549

Description: Weak deduction theorem for eliminating a hypothesis with 2 class variables. Note: if the hypothesis can be separated into two hypotheses, each with one class variable, then dedth2h 4546 is simpler to use. See also comments in dedth 4545. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)

Hypotheses

Ref	Expression
dedth2v.1	⊢ (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓 ↔ 𝜒))
dedth2v.2	⊢ (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒 ↔ 𝜃))
dedth2v.3	⊢ 𝜃

Assertion

Ref	Expression
dedth2v	⊢ (𝜑 → 𝜓)

Proof of Theorem dedth2v

Step	Hyp	Ref	Expression
1		dedth2v.1	. . 3 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓 ↔ 𝜒))
2		dedth2v.2	. . 3 ⊢ (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒 ↔ 𝜃))
3		dedth2v.3	. . 3 ⊢ 𝜃
4	1, 2, 3	dedth2h 4546	. 2 ⊢ ((𝜑 ∧ 𝜑) → 𝜓)
5	4	anidms 576	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1568  ifcif 4486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-if 4487

This theorem is referenced by:  ltweuz  13997  omlsi  31722  pjhfo  32024