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Theorem bnj118 32835
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj118.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj118.2 (𝜑′[1o / 𝑛]𝜑)
Assertion
Ref Expression
bnj118 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
Distinct variable groups:   𝐴,𝑛   𝑅,𝑛   𝑓,𝑛   𝑥,𝑛
Allowed substitution hints:   𝜑(𝑥,𝑓,𝑛)   𝐴(𝑥,𝑓)   𝑅(𝑥,𝑓)   𝜑′(𝑥,𝑓,𝑛)

Proof of Theorem bnj118
StepHypRef Expression
1 bnj118.2 . 2 (𝜑′[1o / 𝑛]𝜑)
2 bnj118.1 . . 3 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
3 bnj105 32689 . . 3 1o ∈ V
42, 3bnj91 32827 . 2 ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
51, 4bitri 274 1 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1539  [wsbc 3716  c0 4257  cfv 6427  1oc1o 8278   predc-bnj14 32653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5222  ax-nul 5229  ax-pow 5287
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3432  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-pw 4536  df-sn 4563  df-suc 6266  df-1o 8285
This theorem is referenced by:  bnj151  32843  bnj153  32846
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