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Theorem bnj1146 31743
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1146.1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Assertion
Ref Expression
bnj1146 𝑥𝐴 𝐵𝐵
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem bnj1146
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfv 1874 . . . . . 6 𝑦(𝑥𝐴𝑤𝐵)
2 bnj1146.1 . . . . . . . 8 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
32nf5i 2085 . . . . . . 7 𝑥 𝑦𝐴
4 nfv 1874 . . . . . . 7 𝑥 𝑤𝐵
53, 4nfan 1863 . . . . . 6 𝑥(𝑦𝐴𝑤𝐵)
6 eleq1w 2841 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
76anbi1d 621 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝐴𝑤𝐵) ↔ (𝑦𝐴𝑤𝐵)))
81, 5, 7cbvexv1 2279 . . . . 5 (∃𝑥(𝑥𝐴𝑤𝐵) ↔ ∃𝑦(𝑦𝐴𝑤𝐵))
9 df-rex 3087 . . . . 5 (∃𝑥𝐴 𝑤𝐵 ↔ ∃𝑥(𝑥𝐴𝑤𝐵))
10 df-rex 3087 . . . . 5 (∃𝑦𝐴 𝑤𝐵 ↔ ∃𝑦(𝑦𝐴𝑤𝐵))
118, 9, 103bitr4i 295 . . . 4 (∃𝑥𝐴 𝑤𝐵 ↔ ∃𝑦𝐴 𝑤𝐵)
1211abbii 2837 . . 3 {𝑤 ∣ ∃𝑥𝐴 𝑤𝐵} = {𝑤 ∣ ∃𝑦𝐴 𝑤𝐵}
13 df-iun 4790 . . 3 𝑥𝐴 𝐵 = {𝑤 ∣ ∃𝑥𝐴 𝑤𝐵}
14 df-iun 4790 . . 3 𝑦𝐴 𝐵 = {𝑤 ∣ ∃𝑦𝐴 𝑤𝐵}
1512, 13, 143eqtr4i 2805 . 2 𝑥𝐴 𝐵 = 𝑦𝐴 𝐵
16 bnj1143 31742 . 2 𝑦𝐴 𝐵𝐵
1715, 16eqsstri 3884 1 𝑥𝐴 𝐵𝐵
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wal 1506  wex 1743  wcel 2051  {cab 2751  wrex 3082  wss 3822   ciun 4788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-v 3410  df-dif 3825  df-in 3829  df-ss 3836  df-nul 4173  df-iun 4790
This theorem is referenced by:  bnj1145  31942
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