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Theorem keephyp2v 4360
Description: Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 4346). (Contributed by NM, 16-Apr-2005.)
Hypotheses
Ref Expression
keephyp2v.1 (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓𝜒))
keephyp2v.2 (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒𝜃))
keephyp2v.3 (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏𝜂))
keephyp2v.4 (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂𝜃))
keephyp2v.5 𝜓
keephyp2v.6 𝜏
Assertion
Ref Expression
keephyp2v 𝜃

Proof of Theorem keephyp2v
StepHypRef Expression
1 keephyp2v.5 . . 3 𝜓
2 iftrue 4296 . . . . . 6 (𝜑 → if(𝜑, 𝐴, 𝐶) = 𝐴)
32eqcomd 2823 . . . . 5 (𝜑𝐴 = if(𝜑, 𝐴, 𝐶))
4 keephyp2v.1 . . . . 5 (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓𝜒))
53, 4syl 17 . . . 4 (𝜑 → (𝜓𝜒))
6 iftrue 4296 . . . . . 6 (𝜑 → if(𝜑, 𝐵, 𝐷) = 𝐵)
76eqcomd 2823 . . . . 5 (𝜑𝐵 = if(𝜑, 𝐵, 𝐷))
8 keephyp2v.2 . . . . 5 (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒𝜃))
97, 8syl 17 . . . 4 (𝜑 → (𝜒𝜃))
105, 9bitrd 270 . . 3 (𝜑 → (𝜓𝜃))
111, 10mpbii 224 . 2 (𝜑𝜃)
12 keephyp2v.6 . . 3 𝜏
13 iffalse 4299 . . . . . 6 𝜑 → if(𝜑, 𝐴, 𝐶) = 𝐶)
1413eqcomd 2823 . . . . 5 𝜑𝐶 = if(𝜑, 𝐴, 𝐶))
15 keephyp2v.3 . . . . 5 (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏𝜂))
1614, 15syl 17 . . . 4 𝜑 → (𝜏𝜂))
17 iffalse 4299 . . . . . 6 𝜑 → if(𝜑, 𝐵, 𝐷) = 𝐷)
1817eqcomd 2823 . . . . 5 𝜑𝐷 = if(𝜑, 𝐵, 𝐷))
19 keephyp2v.4 . . . . 5 (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂𝜃))
2018, 19syl 17 . . . 4 𝜑 → (𝜂𝜃))
2116, 20bitrd 270 . . 3 𝜑 → (𝜏𝜃))
2212, 21mpbii 224 . 2 𝜑𝜃)
2311, 22pm2.61i 176 1 𝜃
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197   = wceq 1637  ifcif 4290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-if 4291
This theorem is referenced by: (None)
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