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Theorem bnj1146 34553
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 1909 . . . . . 6 𝑦(𝑥𝐴𝑤𝐵)
2 bnj1146.1 . . . . . . . 8 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
32nf5i 2134 . . . . . . 7 𝑥 𝑦𝐴
4 nfv 1909 . . . . . . 7 𝑥 𝑤𝐵
53, 4nfan 1894 . . . . . 6 𝑥(𝑦𝐴𝑤𝐵)
6 eleq1w 2808 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
76anbi1d 629 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝐴𝑤𝐵) ↔ (𝑦𝐴𝑤𝐵)))
81, 5, 7cbvexv1 2332 . . . . 5 (∃𝑥(𝑥𝐴𝑤𝐵) ↔ ∃𝑦(𝑦𝐴𝑤𝐵))
9 df-rex 3060 . . . . 5 (∃𝑥𝐴 𝑤𝐵 ↔ ∃𝑥(𝑥𝐴𝑤𝐵))
10 df-rex 3060 . . . . 5 (∃𝑦𝐴 𝑤𝐵 ↔ ∃𝑦(𝑦𝐴𝑤𝐵))
118, 9, 103bitr4i 302 . . . 4 (∃𝑥𝐴 𝑤𝐵 ↔ ∃𝑦𝐴 𝑤𝐵)
1211abbii 2795 . . 3 {𝑤 ∣ ∃𝑥𝐴 𝑤𝐵} = {𝑤 ∣ ∃𝑦𝐴 𝑤𝐵}
13 df-iun 4999 . . 3 𝑥𝐴 𝐵 = {𝑤 ∣ ∃𝑥𝐴 𝑤𝐵}
14 df-iun 4999 . . 3 𝑦𝐴 𝐵 = {𝑤 ∣ ∃𝑦𝐴 𝑤𝐵}
1512, 13, 143eqtr4i 2763 . 2 𝑥𝐴 𝐵 = 𝑦𝐴 𝐵
16 bnj1143 34552 . 2 𝑦𝐴 𝐵𝐵
1715, 16eqsstri 4011 1 𝑥𝐴 𝐵𝐵
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1531  wex 1773  wcel 2098  {cab 2702  wrex 3059  wss 3944   ciun 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-v 3463  df-dif 3947  df-ss 3961  df-nul 4323  df-iun 4999
This theorem is referenced by:  bnj1145  34755
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