Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj609 Structured version   Visualization version   GIF version

Theorem bnj609 31113
Description: Technical lemma for bnj852 31117. 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
bnj609.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj609.2 (𝜑″[𝐺 / 𝑓]𝜑)
bnj609.3 𝐺 ∈ V
Assertion
Ref Expression
bnj609 (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
Distinct variable groups:   𝐴,𝑓   𝑅,𝑓   𝑓,𝑋
Allowed substitution hints:   𝜑(𝑓)   𝐺(𝑓)   𝜑″(𝑓)

Proof of Theorem bnj609
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 bnj609.2 . 2 (𝜑″[𝐺 / 𝑓]𝜑)
2 bnj609.3 . . 3 𝐺 ∈ V
3 dfsbcq 3470 . . 3 (𝑒 = 𝐺 → ([𝑒 / 𝑓]𝜑[𝐺 / 𝑓]𝜑))
4 fveq1 6228 . . . 4 (𝑒 = 𝐺 → (𝑒‘∅) = (𝐺‘∅))
54eqeq1d 2653 . . 3 (𝑒 = 𝐺 → ((𝑒‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)))
6 bnj609.1 . . . . 5 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
76sbcbii 3524 . . . 4 ([𝑒 / 𝑓]𝜑[𝑒 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
8 vex 3234 . . . . 5 𝑒 ∈ V
9 fveq1 6228 . . . . . 6 (𝑓 = 𝑒 → (𝑓‘∅) = (𝑒‘∅))
109eqeq1d 2653 . . . . 5 (𝑓 = 𝑒 → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅)))
118, 10sbcie 3503 . . . 4 ([𝑒 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅))
127, 11bitri 264 . . 3 ([𝑒 / 𝑓]𝜑 ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅))
132, 3, 5, 12vtoclb 3294 . 2 ([𝐺 / 𝑓]𝜑 ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
141, 13bitri 264 1 (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  wcel 2030  Vcvv 3231  [wsbc 3468  c0 3948  cfv 5926   predc-bnj14 30882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-v 3233  df-sbc 3469  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934
This theorem is referenced by:  bnj600  31115  bnj908  31127  bnj934  31131
  Copyright terms: Public domain W3C validator