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Theorem isarep1 6155
Description: Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by 𝜑(𝑥, 𝑦) i.e. the class ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴). If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
Assertion
Ref Expression
isarep1 (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥𝐴 [𝑏 / 𝑦]𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑏,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑏)   𝐴(𝑦,𝑏)

Proof of Theorem isarep1
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 vex 3353 . . 3 𝑏 ∈ V
21elima 5653 . 2 (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑧𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏)
3 df-br 4810 . . . 4 (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4 opelopabsb 5146 . . . 4 (⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
5 sbsbc 3600 . . . . . 6 ([𝑏 / 𝑦]𝜑[𝑏 / 𝑦]𝜑)
65sbbii 2068 . . . . 5 ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
7 sbsbc 3600 . . . . 5 ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑[𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
86, 7bitr2i 267 . . . 4 ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
93, 4, 83bitri 288 . . 3 (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
109rexbii 3188 . 2 (∃𝑧𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ∃𝑧𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
11 nfs1v 2287 . . 3 𝑥[𝑧 / 𝑥][𝑏 / 𝑦]𝜑
12 nfv 2009 . . 3 𝑧[𝑏 / 𝑦]𝜑
13 sbequ12r 2279 . . 3 (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
1411, 12, 13cbvrex 3316 . 2 (∃𝑧𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ ∃𝑥𝐴 [𝑏 / 𝑦]𝜑)
152, 10, 143bitri 288 1 (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥𝐴 [𝑏 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 197  [wsb 2061  wcel 2155  wrex 3056  [wsbc 3596  cop 4340   class class class wbr 4809  {copab 4871  cima 5280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-xp 5283  df-cnv 5285  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290
This theorem is referenced by: (None)
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