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Theorem isarep1 6634
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.) (Proof shortened by SN, 19-Dec-2024.)
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 3478 . . 3 𝑏 ∈ V
21elima 6062 . 2 (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑧𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏)
3 df-br 5148 . . . 4 (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4 vopelopabsb 5528 . . . 4 (⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
53, 4bitri 274 . . 3 (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
65rexbii 3094 . 2 (∃𝑧𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ∃𝑧𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
7 nfs1v 2153 . . 3 𝑥[𝑧 / 𝑥][𝑏 / 𝑦]𝜑
8 nfv 1917 . . 3 𝑧[𝑏 / 𝑦]𝜑
9 sbequ12r 2244 . . 3 (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
107, 8, 9cbvrexw 3304 . 2 (∃𝑧𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ ∃𝑥𝐴 [𝑏 / 𝑦]𝜑)
112, 6, 103bitri 296 1 (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥𝐴 [𝑏 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  [wsb 2067  wcel 2106  wrex 3070  cop 4633   class class class wbr 5147  {copab 5209  cima 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688
This theorem is referenced by: (None)
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