Theorem dedth2v 4555

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 4552 is simpler to use. See also comments in dedth 4551. (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 4552	. 2 ⊢ ((𝜑 ∧ 𝜑) → 𝜓)
5	4	anidms 577	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ifcif 4492

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-if 4493

This theorem is used by:  ltweuz  14017  omlsi  31786  pjhfo  32088