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Theorem isarep1 5256
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 2715 . . 3 𝑏 ∈ V
21elima 4933 . 2 (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑧𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏)
3 df-br 3966 . . . 4 (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4 opelopabsb 4220 . . . 4 (⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
5 sbsbc 2941 . . . . . 6 ([𝑏 / 𝑦]𝜑[𝑏 / 𝑦]𝜑)
65sbbii 1745 . . . . 5 ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
7 sbsbc 2941 . . . . 5 ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑[𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
86, 7bitr2i 184 . . . 4 ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
93, 4, 83bitri 205 . . 3 (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
109rexbii 2464 . 2 (∃𝑧𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ∃𝑧𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
11 nfs1v 1919 . . 3 𝑥[𝑧 / 𝑥][𝑏 / 𝑦]𝜑
12 nfv 1508 . . 3 𝑧[𝑏 / 𝑦]𝜑
13 sbequ12r 1752 . . 3 (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
1411, 12, 13cbvrex 2677 . 2 (∃𝑧𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ ∃𝑥𝐴 [𝑏 / 𝑦]𝜑)
152, 10, 143bitri 205 1 (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥𝐴 [𝑏 / 𝑦]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 104  [wsb 1742  wcel 2128  wrex 2436  [wsbc 2937  cop 3563   class class class wbr 3965  {copab 4024  cima 4589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4592  df-cnv 4594  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599
This theorem is referenced by: (None)
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