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Theorem nfopab1 5213
Description: The first abstraction variable in an ordered-pair class abstraction is effectively not free. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
nfopab1 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Proof of Theorem nfopab1
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-opab 5206 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfe1 2150 . . 3 𝑥𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
32nfab 2911 . 2 𝑥{𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
41, 3nfcxfr 2903 1 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wex 1779  {cab 2714  wnfc 2890  cop 4632  {copab 5205
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-opab 5206
This theorem is referenced by:  nfmpt1  5250  rexopabb  5533  ssopab2bw  5552  ssopab2b  5554  dmopab  5926  rnopab  5965  funopab  6601  fvopab5  7049  zfrep6  7979  opabdm  32623  opabrn  32624  fpwrelmap  32744  fineqvrep  35109  bj-opabco  37189  vvdifopab  38261  aomclem8  43073  sprsymrelf  47482
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