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Theorem bnj1146 30116
 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 1830 . . . . . 6 𝑦(𝑥𝐴𝑤𝐵)
2 bnj1146.1 . . . . . . . 8 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
32nf5i 2011 . . . . . . 7 𝑥 𝑦𝐴
4 nfv 1830 . . . . . . 7 𝑥 𝑤𝐵
53, 4nfan 1816 . . . . . 6 𝑥(𝑦𝐴𝑤𝐵)
6 eleq1 2676 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
76anbi1d 737 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝐴𝑤𝐵) ↔ (𝑦𝐴𝑤𝐵)))
81, 5, 7cbvex 2260 . . . . 5 (∃𝑥(𝑥𝐴𝑤𝐵) ↔ ∃𝑦(𝑦𝐴𝑤𝐵))
9 df-rex 2902 . . . . 5 (∃𝑥𝐴 𝑤𝐵 ↔ ∃𝑥(𝑥𝐴𝑤𝐵))
10 df-rex 2902 . . . . 5 (∃𝑦𝐴 𝑤𝐵 ↔ ∃𝑦(𝑦𝐴𝑤𝐵))
118, 9, 103bitr4i 291 . . . 4 (∃𝑥𝐴 𝑤𝐵 ↔ ∃𝑦𝐴 𝑤𝐵)
1211abbii 2726 . . 3 {𝑤 ∣ ∃𝑥𝐴 𝑤𝐵} = {𝑤 ∣ ∃𝑦𝐴 𝑤𝐵}
13 df-iun 4457 . . 3 𝑥𝐴 𝐵 = {𝑤 ∣ ∃𝑥𝐴 𝑤𝐵}
14 df-iun 4457 . . 3 𝑦𝐴 𝐵 = {𝑤 ∣ ∃𝑦𝐴 𝑤𝐵}
1512, 13, 143eqtr4i 2642 . 2 𝑥𝐴 𝐵 = 𝑦𝐴 𝐵
16 bnj1143 30115 . 2 𝑦𝐴 𝐵𝐵
1715, 16eqsstri 3598 1 𝑥𝐴 𝐵𝐵
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∃wrex 2897   ⊆ wss 3540  ∪ ciun 4455 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-iun 4457 This theorem is referenced by:  bnj1145  30315
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