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Theorem bnj92 34626
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj92.1 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj92.2 𝑍 ∈ V
Assertion
Ref Expression
bnj92 ([𝑍 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑍 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
Distinct variable groups:   𝐴,𝑛   𝑅,𝑛   𝑖,𝑍   𝑓,𝑛   𝑖,𝑛   𝑦,𝑛
Allowed substitution hints:   𝜓(𝑦,𝑓,𝑖,𝑛)   𝐴(𝑦,𝑓,𝑖)   𝑅(𝑦,𝑓,𝑖)   𝑍(𝑦,𝑓,𝑛)

Proof of Theorem bnj92
StepHypRef Expression
1 bnj92.1 . . 3 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
21sbcbii 3834 . 2 ([𝑍 / 𝑛]𝜓[𝑍 / 𝑛]𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj92.2 . . 3 𝑍 ∈ V
43bnj538 34504 . 2 ([𝑍 / 𝑛]𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω [𝑍 / 𝑛](suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
5 sbcimg 3825 . . . . 5 (𝑍 ∈ V → ([𝑍 / 𝑛](suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ([𝑍 / 𝑛]suc 𝑖𝑛[𝑍 / 𝑛](𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
63, 5ax-mp 5 . . . 4 ([𝑍 / 𝑛](suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ([𝑍 / 𝑛]suc 𝑖𝑛[𝑍 / 𝑛](𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
7 sbcel2gv 3845 . . . . . 6 (𝑍 ∈ V → ([𝑍 / 𝑛]suc 𝑖𝑛 ↔ suc 𝑖𝑍))
83, 7ax-mp 5 . . . . 5 ([𝑍 / 𝑛]suc 𝑖𝑛 ↔ suc 𝑖𝑍)
93bnj525 34502 . . . . 5 ([𝑍 / 𝑛](𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
108, 9imbi12i 349 . . . 4 (([𝑍 / 𝑛]suc 𝑖𝑛[𝑍 / 𝑛](𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖𝑍 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
116, 10bitri 274 . . 3 ([𝑍 / 𝑛](suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖𝑍 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
1211ralbii 3082 . 2 (∀𝑖 ∈ ω [𝑍 / 𝑛](suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖𝑍 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
132, 4, 123bitri 296 1 ([𝑍 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑍 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  wral 3050  Vcvv 3461  [wsbc 3773   ciun 4997  suc csuc 6373  cfv 6549  ωcom 7871   predc-bnj14 34452
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-v 3463  df-sbc 3774
This theorem is referenced by:  bnj106  34632  bnj153  34644
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