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Theorem bnj1146 34784
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 1912 . . . . . 6 𝑦(𝑥𝐴𝑤𝐵)
2 bnj1146.1 . . . . . . . 8 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
32nf5i 2144 . . . . . . 7 𝑥 𝑦𝐴
4 nfv 1912 . . . . . . 7 𝑥 𝑤𝐵
53, 4nfan 1897 . . . . . 6 𝑥(𝑦𝐴𝑤𝐵)
6 eleq1w 2822 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
76anbi1d 631 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝐴𝑤𝐵) ↔ (𝑦𝐴𝑤𝐵)))
81, 5, 7cbvexv1 2343 . . . . 5 (∃𝑥(𝑥𝐴𝑤𝐵) ↔ ∃𝑦(𝑦𝐴𝑤𝐵))
9 df-rex 3069 . . . . 5 (∃𝑥𝐴 𝑤𝐵 ↔ ∃𝑥(𝑥𝐴𝑤𝐵))
10 df-rex 3069 . . . . 5 (∃𝑦𝐴 𝑤𝐵 ↔ ∃𝑦(𝑦𝐴𝑤𝐵))
118, 9, 103bitr4i 303 . . . 4 (∃𝑥𝐴 𝑤𝐵 ↔ ∃𝑦𝐴 𝑤𝐵)
1211abbii 2807 . . 3 {𝑤 ∣ ∃𝑥𝐴 𝑤𝐵} = {𝑤 ∣ ∃𝑦𝐴 𝑤𝐵}
13 df-iun 4998 . . 3 𝑥𝐴 𝐵 = {𝑤 ∣ ∃𝑥𝐴 𝑤𝐵}
14 df-iun 4998 . . 3 𝑦𝐴 𝐵 = {𝑤 ∣ ∃𝑦𝐴 𝑤𝐵}
1512, 13, 143eqtr4i 2773 . 2 𝑥𝐴 𝐵 = 𝑦𝐴 𝐵
16 bnj1143 34783 . 2 𝑦𝐴 𝐵𝐵
1715, 16eqsstri 4030 1 𝑥𝐴 𝐵𝐵
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535  wex 1776  wcel 2106  {cab 2712  wrex 3068  wss 3963   ciun 4996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-v 3480  df-dif 3966  df-ss 3980  df-nul 4340  df-iun 4998
This theorem is referenced by:  bnj1145  34986
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