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Theorem bnj118 32201
 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 32054 . . 3 1o ∈ V
42, 3bnj91 32193 . 2 ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
51, 4bitri 278 1 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538  [wsbc 3759  ∅c0 4277  ‘cfv 6344  1oc1o 8092   predc-bnj14 32018 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 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  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 3483  df-sbc 3760  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-pw 4525  df-sn 4552  df-suc 6185  df-1o 8099 This theorem is referenced by:  bnj151  32209  bnj153  32212
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