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Theorem bnj125 31280
Description: Technical lemma for bnj150 31284. 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 3643 . . 3 ([𝐹 / 𝑓]𝜑′[𝐹 / 𝑓][1𝑜 / 𝑛]𝜑)
4 bnj125.1 . . . . . 6 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
5 bnj105 31130 . . . . . 6 1𝑜 ∈ V
64, 5bnj91 31269 . . . . 5 ([1𝑜 / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
76sbcbii 3643 . . . 4 ([𝐹 / 𝑓][1𝑜 / 𝑛]𝜑[𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
8 bnj125.4 . . . . . 6 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
98bnj95 31272 . . . . 5 𝐹 ∈ V
10 fveq1 6331 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘∅) = (𝐹‘∅))
1110eqeq1d 2773 . . . . 5 (𝑓 = 𝐹 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)))
129, 11sbcie 3622 . . . 4 ([𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
137, 12bitri 264 . . 3 ([𝐹 / 𝑓][1𝑜 / 𝑛]𝜑 ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
143, 13bitri 264 . 2 ([𝐹 / 𝑓]𝜑′ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
151, 14bitri 264 1 (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  [wsbc 3587  c0 4063  {csn 4316  cop 4322  cfv 6031  1𝑜c1o 7706   predc-bnj14 31094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-pw 4299  df-sn 4317  df-pr 4319  df-uni 4575  df-br 4787  df-suc 5872  df-iota 5994  df-fv 6039  df-1o 7713
This theorem is referenced by:  bnj150  31284  bnj153  31288
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