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Theorem bnj91 32022
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj91.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj91.2 𝑍 ∈ V
Assertion
Ref Expression
bnj91 ([𝑍 / 𝑦]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑅   𝑦,𝑓   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓)   𝐴(𝑥,𝑓)   𝑅(𝑥,𝑓)   𝑍(𝑥,𝑦,𝑓)

Proof of Theorem bnj91
StepHypRef Expression
1 bnj91.1 . . 3 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
21sbcbii 3833 . 2 ([𝑍 / 𝑦]𝜑[𝑍 / 𝑦](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
3 bnj91.2 . . 3 𝑍 ∈ V
43bnj525 31898 . 2 ([𝑍 / 𝑦](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
52, 4bitri 276 1 ([𝑍 / 𝑦]𝜑 ↔ (𝑓‘∅) = 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 6352   predc-bnj14 31847
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-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-sbc 3777
This theorem is referenced by:  bnj118  32030  bnj125  32033
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