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Theorem isarep1 5935
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 3189 . . 3 𝑏 ∈ V
21elima 5430 . 2 (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑧𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏)
3 df-br 4614 . . . 4 (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4 opelopabsb 4945 . . . 4 (⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
5 sbsbc 3421 . . . . . 6 ([𝑏 / 𝑦]𝜑[𝑏 / 𝑦]𝜑)
65sbbii 1884 . . . . 5 ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
7 sbsbc 3421 . . . . 5 ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑[𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
86, 7bitr2i 265 . . . 4 ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
93, 4, 83bitri 286 . . 3 (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
109rexbii 3034 . 2 (∃𝑧𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ∃𝑧𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
11 nfs1v 2436 . . 3 𝑥[𝑧 / 𝑥][𝑏 / 𝑦]𝜑
12 nfv 1840 . . 3 𝑧[𝑏 / 𝑦]𝜑
13 sbequ12r 2109 . . 3 (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
1411, 12, 13cbvrex 3156 . 2 (∃𝑧𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ ∃𝑥𝐴 [𝑏 / 𝑦]𝜑)
152, 10, 143bitri 286 1 (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥𝐴 [𝑏 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  [wsb 1877  wcel 1987  wrex 2908  [wsbc 3417  cop 4154   class class class wbr 4613  {copab 4672  cima 5077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087
This theorem is referenced by: (None)
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