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Theorem bnj539 31478
Description: Technical lemma for bnj852 31508. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj539.1 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj539.2 (𝜓′[𝑀 / 𝑛]𝜓)
bnj539.3 𝑀 ∈ V
Assertion
Ref Expression
bnj539 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑀 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
Distinct variable groups:   𝐴,𝑛   𝑛,𝐹   𝑖,𝑀   𝑅,𝑛   𝑖,𝑛   𝑦,𝑛
Allowed substitution hints:   𝜓(𝑦,𝑖,𝑛)   𝐴(𝑦,𝑖)   𝑅(𝑦,𝑖)   𝐹(𝑦,𝑖)   𝑀(𝑦,𝑛)   𝜓′(𝑦,𝑖,𝑛)

Proof of Theorem bnj539
StepHypRef Expression
1 bnj539.2 . 2 (𝜓′[𝑀 / 𝑛]𝜓)
2 bnj539.1 . . . 4 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
32sbcbii 3689 . . 3 ([𝑀 / 𝑛]𝜓[𝑀 / 𝑛]𝑖 ∈ ω (suc 𝑖𝑛 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
4 bnj539.3 . . . . 5 𝑀 ∈ V
54bnj538 31327 . . . 4 ([𝑀 / 𝑛]𝑖 ∈ ω (suc 𝑖𝑛 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω [𝑀 / 𝑛](suc 𝑖𝑛 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
6 sbcimg 3675 . . . . . . 7 (𝑀 ∈ V → ([𝑀 / 𝑛](suc 𝑖𝑛 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ([𝑀 / 𝑛]suc 𝑖𝑛[𝑀 / 𝑛](𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))))
74, 6ax-mp 5 . . . . . 6 ([𝑀 / 𝑛](suc 𝑖𝑛 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ([𝑀 / 𝑛]suc 𝑖𝑛[𝑀 / 𝑛](𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
8 sbcel2gv 3693 . . . . . . . 8 (𝑀 ∈ V → ([𝑀 / 𝑛]suc 𝑖𝑛 ↔ suc 𝑖𝑀))
94, 8ax-mp 5 . . . . . . 7 ([𝑀 / 𝑛]suc 𝑖𝑛 ↔ suc 𝑖𝑀)
104bnj525 31325 . . . . . . 7 ([𝑀 / 𝑛](𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))
119, 10imbi12i 342 . . . . . 6 (([𝑀 / 𝑛]suc 𝑖𝑛[𝑀 / 𝑛](𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖𝑀 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
127, 11bitri 267 . . . . 5 ([𝑀 / 𝑛](suc 𝑖𝑛 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖𝑀 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
1312ralbii 3161 . . . 4 (∀𝑖 ∈ ω [𝑀 / 𝑛](suc 𝑖𝑛 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖𝑀 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
145, 13bitri 267 . . 3 ([𝑀 / 𝑛]𝑖 ∈ ω (suc 𝑖𝑛 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑖 ∈ ω (suc 𝑖𝑀 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
153, 14bitri 267 . 2 ([𝑀 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑀 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
161, 15bitri 267 1 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑀 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wcel 2157  wral 3089  Vcvv 3385  [wsbc 3633   ciun 4710  suc csuc 5943  cfv 6101  ωcom 7299   predc-bnj14 31274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-v 3387  df-sbc 3634
This theorem is referenced by:  bnj600  31506  bnj908  31518  bnj964  31530  bnj999  31544
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