Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj934 Structured version   Visualization version   GIF version

Theorem bnj934 31522
Description: Technical lemma for bnj69 31595. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj934.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj934.4 (𝜑′[𝑝 / 𝑛]𝜑)
bnj934.7 (𝜑″[𝐺 / 𝑓]𝜑′)
bnj934.50 𝐺 ∈ V
Assertion
Ref Expression
bnj934 ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″)
Distinct variable groups:   𝐴,𝑓,𝑛   𝑅,𝑓,𝑛   𝑓,𝑋,𝑛
Allowed substitution hints:   𝜑(𝑓,𝑛,𝑝)   𝐴(𝑝)   𝑅(𝑝)   𝐺(𝑓,𝑛,𝑝)   𝑋(𝑝)   𝜑′(𝑓,𝑛,𝑝)   𝜑″(𝑓,𝑛,𝑝)

Proof of Theorem bnj934
StepHypRef Expression
1 bnj934.1 . . . 4 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2 eqtr 2818 . . . 4 (((𝐺‘∅) = (𝑓‘∅) ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) → (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
31, 2sylan2b 588 . . 3 (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑) → (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
4 bnj934.7 . . . . 5 (𝜑″[𝐺 / 𝑓]𝜑′)
5 bnj934.4 . . . . . . . 8 (𝜑′[𝑝 / 𝑛]𝜑)
6 vex 3388 . . . . . . . 8 𝑝 ∈ V
71, 5, 6bnj523 31474 . . . . . . 7 (𝜑′ ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
87, 1bitr4i 270 . . . . . 6 (𝜑′𝜑)
98sbcbii 3689 . . . . 5 ([𝐺 / 𝑓]𝜑′[𝐺 / 𝑓]𝜑)
104, 9bitri 267 . . . 4 (𝜑″[𝐺 / 𝑓]𝜑)
11 bnj934.50 . . . 4 𝐺 ∈ V
121, 10, 11bnj609 31504 . . 3 (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
133, 12sylibr 226 . 2 (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑) → 𝜑″)
1413ancoms 451 1 ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  [wsbc 3633  c0 4115  cfv 6101   predc-bnj14 31274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-v 3387  df-sbc 3634  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109
This theorem is referenced by:  bnj929  31523
  Copyright terms: Public domain W3C validator