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Theorem isarep1OLD 6657
Description: Obsolete version of isarep1 6656 as of 19-Dec-2024. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
isarep1OLD (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 isarep1OLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 vex 3484 . . 3 𝑏 ∈ V
21elima 6083 . 2 (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑧𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏)
3 df-br 5144 . . . 4 (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4 opelopabsb 5535 . . . 4 (⟨𝑧, 𝑏⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
5 sbsbc 3792 . . . . . 6 ([𝑏 / 𝑦]𝜑[𝑏 / 𝑦]𝜑)
65sbbii 2076 . . . . 5 ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
7 sbsbc 3792 . . . . 5 ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑[𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
86, 7bitr2i 276 . . . 4 ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
93, 4, 83bitri 297 . . 3 (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
109rexbii 3094 . 2 (∃𝑧𝐴 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑏 ↔ ∃𝑧𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑)
11 nfs1v 2156 . . 3 𝑥[𝑧 / 𝑥][𝑏 / 𝑦]𝜑
12 nfv 1914 . . 3 𝑧[𝑏 / 𝑦]𝜑
13 sbequ12r 2252 . . 3 (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
1411, 12, 13cbvrexw 3307 . 2 (∃𝑧𝐴 [𝑧 / 𝑥][𝑏 / 𝑦]𝜑 ↔ ∃𝑥𝐴 [𝑏 / 𝑦]𝜑)
152, 10, 143bitri 297 1 (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥𝐴 [𝑏 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  [wsb 2064  wcel 2108  wrex 3070  [wsbc 3788  cop 4632   class class class wbr 5143  {copab 5205  cima 5688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by: (None)
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