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Theorem bnj92 34855
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 3852 . 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 34733 . 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 3843 . . . . 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 3863 . . . . . 6 (𝑍 ∈ V → ([𝑍 / 𝑛]suc 𝑖𝑛 ↔ suc 𝑖𝑍))
83, 7ax-mp 5 . . . . 5 ([𝑍 / 𝑛]suc 𝑖𝑛 ↔ suc 𝑖𝑍)
93bnj525 34731 . . . . 5 ([𝑍 / 𝑛](𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
108, 9imbi12i 350 . . . 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 275 . . 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 3091 . 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 297 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 206   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  [wsbc 3791   ciun 4996  suc csuc 6388  cfv 6563  ωcom 7887   predc-bnj14 34681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-v 3480  df-sbc 3792
This theorem is referenced by:  bnj106  34861  bnj153  34873
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