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Theorem elimhyp 3710
 Description: Eliminate a hypothesis containing class variable A when it is known for a specific class B. For more information, see comments in dedth 3703. (Contributed by NM, 15-May-1999.)
Hypotheses
Ref Expression
elimhyp.1 (A = if(φ, A, B) → (φψ))
elimhyp.2 (B = if(φ, A, B) → (χψ))
elimhyp.3 χ
Assertion
Ref Expression
elimhyp ψ

Proof of Theorem elimhyp
StepHypRef Expression
1 iftrue 3668 . . . . 5 (φ → if(φ, A, B) = A)
21eqcomd 2358 . . . 4 (φA = if(φ, A, B))
3 elimhyp.1 . . . 4 (A = if(φ, A, B) → (φψ))
42, 3syl 15 . . 3 (φ → (φψ))
54ibi 232 . 2 (φψ)
6 elimhyp.3 . . 3 χ
7 iffalse 3669 . . . . 5 φ → if(φ, A, B) = B)
87eqcomd 2358 . . . 4 φB = if(φ, A, B))
9 elimhyp.2 . . . 4 (B = if(φ, A, B) → (χψ))
108, 9syl 15 . . 3 φ → (χψ))
116, 10mpbii 202 . 2 φψ)
125, 11pm2.61i 156 1 ψ
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   = wceq 1642   ifcif 3662 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3663 This theorem is referenced by:  elimel  3714  elimf  5222
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