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Theorem bnj609 31504
Description: Technical lemma for bnj852 31508. 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 3635 . . 3 (𝑒 = 𝐺 → ([𝑒 / 𝑓]𝜑[𝐺 / 𝑓]𝜑))
4 fveq1 6410 . . . 4 (𝑒 = 𝐺 → (𝑒‘∅) = (𝐺‘∅))
54eqeq1d 2801 . . 3 (𝑒 = 𝐺 → ((𝑒‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)))
6 bnj609.1 . . . . 5 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
76sbcbii 3689 . . . 4 ([𝑒 / 𝑓]𝜑[𝑒 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
8 vex 3388 . . . . 5 𝑒 ∈ V
9 fveq1 6410 . . . . . 6 (𝑓 = 𝑒 → (𝑓‘∅) = (𝑒‘∅))
109eqeq1d 2801 . . . . 5 (𝑓 = 𝑒 → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅)))
118, 10sbcie 3668 . . . 4 ([𝑒 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅))
127, 11bitri 267 . . 3 ([𝑒 / 𝑓]𝜑 ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅))
132, 3, 5, 12vtoclb 3450 . 2 ([𝐺 / 𝑓]𝜑 ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
141, 13bitri 267 1 (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 198   = 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-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:  bnj600  31506  bnj908  31518  bnj934  31522
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