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Theorem bnj539 32865
Description: Technical lemma for bnj852 32895. 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 3781 . . 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 32714 . . . 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 3771 . . . . . . 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 3793 . . . . . . . 8 (𝑀 ∈ V → ([𝑀 / 𝑛]suc 𝑖𝑛 ↔ suc 𝑖𝑀))
94, 8ax-mp 5 . . . . . . 7 ([𝑀 / 𝑛]suc 𝑖𝑛 ↔ suc 𝑖𝑀)
104bnj525 32712 . . . . . . 7 ([𝑀 / 𝑛](𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅))
119, 10imbi12i 351 . . . . . 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 274 . . . . 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 3093 . . . 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 274 . . 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 274 . 2 ([𝑀 / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑀 → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
161, 15bitri 274 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 205   = wceq 1542  wcel 2110  wral 3066  Vcvv 3431  [wsbc 3720   ciun 4930  suc csuc 6266  cfv 6431  ωcom 7704   predc-bnj14 32661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-v 3433  df-sbc 3721
This theorem is referenced by:  bnj600  32893  bnj908  32905  bnj964  32917  bnj999  32932
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