Theorem dedth2v 4587

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 4584 is simpler to use. See also comments in dedth 4583. (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 4584	. 2 ⊢ ((𝜑 ∧ 𝜑) → 𝜓)
5	4	anidms 566	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539  ifcif 4524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-if 4525

This theorem is referenced by:  ltweuz  14003  omlsi  31424  pjhfo  31726