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Theorem elimdhyp 3715
 Description: Version of elimhyp 3710 where the hypothesis is deduced from the final antecedent. See ghomgrplem in set.mm for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)
Hypotheses
Ref Expression
elimdhyp.1 (φψ)
elimdhyp.2 (A = if(φ, A, B) → (ψχ))
elimdhyp.3 (B = if(φ, A, B) → (θχ))
elimdhyp.4 θ
Assertion
Ref Expression
elimdhyp χ

Proof of Theorem elimdhyp
StepHypRef Expression
1 elimdhyp.1 . . 3 (φψ)
2 iftrue 3668 . . . . 5 (φ → if(φ, A, B) = A)
32eqcomd 2358 . . . 4 (φA = if(φ, A, B))
4 elimdhyp.2 . . . 4 (A = if(φ, A, B) → (ψχ))
53, 4syl 15 . . 3 (φ → (ψχ))
61, 5mpbid 201 . 2 (φχ)
7 elimdhyp.4 . . 3 θ
8 iffalse 3669 . . . . 5 φ → if(φ, A, B) = B)
98eqcomd 2358 . . . 4 φB = if(φ, A, B))
10 elimdhyp.3 . . . 4 (B = if(φ, A, B) → (θχ))
119, 10syl 15 . . 3 φ → (θχ))
127, 11mpbii 202 . 2 φχ)
136, 12pm2.61i 156 1 χ
 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-mp 5  ax-1 6  ax-2 7  ax-3 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: (None)
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