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Theorem bnj125 32229
Description: Technical lemma for bnj150 32233. 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
bnj125.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj125.2 (𝜑′[1o / 𝑛]𝜑)
bnj125.3 (𝜑″[𝐹 / 𝑓]𝜑′)
bnj125.4 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
Assertion
Ref Expression
bnj125 (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
Distinct variable groups:   𝐴,𝑓,𝑛   𝑓,𝐹   𝑅,𝑓,𝑛   𝑥,𝑓,𝑛
Allowed substitution hints:   𝜑(𝑥,𝑓,𝑛)   𝐴(𝑥)   𝑅(𝑥)   𝐹(𝑥,𝑛)   𝜑′(𝑥,𝑓,𝑛)   𝜑″(𝑥,𝑓,𝑛)

Proof of Theorem bnj125
StepHypRef Expression
1 bnj125.3 . 2 (𝜑″[𝐹 / 𝑓]𝜑′)
2 bnj125.2 . . . 4 (𝜑′[1o / 𝑛]𝜑)
32sbcbii 3814 . . 3 ([𝐹 / 𝑓]𝜑′[𝐹 / 𝑓][1o / 𝑛]𝜑)
4 bnj125.1 . . . . . 6 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
5 bnj105 32079 . . . . . 6 1o ∈ V
64, 5bnj91 32218 . . . . 5 ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
76sbcbii 3814 . . . 4 ([𝐹 / 𝑓][1o / 𝑛]𝜑[𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
8 bnj125.4 . . . . . 6 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
98bnj95 32221 . . . . 5 𝐹 ∈ V
10 fveq1 6662 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘∅) = (𝐹‘∅))
1110eqeq1d 2826 . . . . 5 (𝑓 = 𝐹 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)))
129, 11sbcie 3798 . . . 4 ([𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
137, 12bitri 278 . . 3 ([𝐹 / 𝑓][1o / 𝑛]𝜑 ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
143, 13bitri 278 . 2 ([𝐹 / 𝑓]𝜑′ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
151, 14bitri 278 1 (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  [wsbc 3758  c0 4276  {csn 4550  cop 4556  cfv 6345  1oc1o 8093   predc-bnj14 32043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-pw 4524  df-sn 4551  df-pr 4553  df-uni 4825  df-br 5054  df-suc 6186  df-iota 6304  df-fv 6353  df-1o 8100
This theorem is referenced by:  bnj150  32233  bnj153  32237
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