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Theorem bnj91 32365
 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 3755 . 2 ([𝑍 / 𝑦]𝜑[𝑍 / 𝑦](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
3 bnj91.2 . . 3 𝑍 ∈ V
43bnj525 32241 . 2 ([𝑍 / 𝑦](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
52, 4bitri 278 1 ([𝑍 / 𝑦]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Vcvv 3409  [wsbc 3698  ∅c0 4227  ‘cfv 6339   predc-bnj14 32190 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-sbc 3699 This theorem is referenced by:  bnj118  32373  bnj125  32376
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