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Theorem bnj609 32094
 Description: Technical lemma for bnj852 32098. 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 3778 . . 3 (𝑒 = 𝐺 → ([𝑒 / 𝑓]𝜑[𝐺 / 𝑓]𝜑))
4 fveq1 6668 . . . 4 (𝑒 = 𝐺 → (𝑒‘∅) = (𝐺‘∅))
54eqeq1d 2828 . . 3 (𝑒 = 𝐺 → ((𝑒‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅)))
6 bnj609.1 . . . . 5 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
76sbcbii 3833 . . . 4 ([𝑒 / 𝑓]𝜑[𝑒 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
8 vex 3503 . . . . 5 𝑒 ∈ V
9 fveq1 6668 . . . . . 6 (𝑓 = 𝑒 → (𝑓‘∅) = (𝑒‘∅))
109eqeq1d 2828 . . . . 5 (𝑓 = 𝑒 → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅)))
118, 10sbcie 3816 . . . 4 ([𝑒 / 𝑓](𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅))
127, 11bitri 276 . . 3 ([𝑒 / 𝑓]𝜑 ↔ (𝑒‘∅) = pred(𝑋, 𝐴, 𝑅))
132, 3, 5, 12vtoclb 3570 . 2 ([𝐺 / 𝑓]𝜑 ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
141, 13bitri 276 1 (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   = wceq 1530   ∈ wcel 2107  Vcvv 3500  [wsbc 3776  ∅c0 4295  ‘cfv 6354   predc-bnj14 31863 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2169  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-rex 3149  df-v 3502  df-sbc 3777  df-uni 4838  df-br 5064  df-iota 6313  df-fv 6362 This theorem is referenced by:  bnj600  32096  bnj908  32108  bnj934  32112
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