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Theorem bnj91 34875
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 3846 . 2 ([𝑍 / 𝑦]𝜑[𝑍 / 𝑦](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
3 bnj91.2 . . 3 𝑍 ∈ V
43bnj525 34752 . 2 ([𝑍 / 𝑦](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
52, 4bitri 275 1 ([𝑍 / 𝑦]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wcel 2108  Vcvv 3480  [wsbc 3788  c0 4333  cfv 6561   predc-bnj14 34702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-sbc 3789
This theorem is referenced by:  bnj118  34883  bnj125  34886
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