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Theorem bnj106 32561
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj106.1 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj106.2 𝐹 ∈ V
Assertion
Ref Expression
bnj106 ([𝐹 / 𝑓][1o / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
Distinct variable groups:   𝐴,𝑓,𝑛   𝑓,𝐹,𝑖,𝑦   𝑅,𝑓,𝑛   𝑖,𝑛,𝑦
Allowed substitution hints:   𝜓(𝑦,𝑓,𝑖,𝑛)   𝐴(𝑦,𝑖)   𝑅(𝑦,𝑖)   𝐹(𝑛)

Proof of Theorem bnj106
StepHypRef Expression
1 bnj106.1 . . . 4 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
2 bnj105 32415 . . . 4 1o ∈ V
31, 2bnj92 32555 . . 3 ([1o / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
43sbcbii 3755 . 2 ([𝐹 / 𝑓][1o / 𝑛]𝜓[𝐹 / 𝑓]𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
5 bnj106.2 . . 3 𝐹 ∈ V
6 fveq1 6716 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘suc 𝑖) = (𝐹‘suc 𝑖))
7 fveq1 6716 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑖) = (𝐹𝑖))
87bnj1113 32478 . . . . . 6 (𝑓 = 𝐹 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))
96, 8eqeq12d 2753 . . . . 5 (𝑓 = 𝐹 → ((𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
109imbi2d 344 . . . 4 (𝑓 = 𝐹 → ((suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))))
1110ralbidv 3118 . . 3 (𝑓 = 𝐹 → (∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))))
125, 11sbcie 3737 . 2 ([𝐹 / 𝑓]𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
134, 12bitri 278 1 ([𝐹 / 𝑓][1o / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wcel 2110  wral 3061  Vcvv 3408  [wsbc 3694   ciun 4904  suc csuc 6215  cfv 6380  ωcom 7644  1oc1o 8195   predc-bnj14 32379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-pw 4515  df-sn 4542  df-uni 4820  df-iun 4906  df-br 5054  df-suc 6219  df-iota 6338  df-fv 6388  df-1o 8202
This theorem is referenced by:  bnj126  32566
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