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Theorem bnj1146 29922
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 1829 . . . . . 6 𝑦(𝑥𝐴𝑤𝐵)
2 bnj1146.1 . . . . . . . 8 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
32nf5i 2010 . . . . . . 7 𝑥 𝑦𝐴
4 nfv 1829 . . . . . . 7 𝑥 𝑤𝐵
53, 4nfan 1815 . . . . . 6 𝑥(𝑦𝐴𝑤𝐵)
6 eleq1 2675 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
76anbi1d 736 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝐴𝑤𝐵) ↔ (𝑦𝐴𝑤𝐵)))
81, 5, 7cbvex 2258 . . . . 5 (∃𝑥(𝑥𝐴𝑤𝐵) ↔ ∃𝑦(𝑦𝐴𝑤𝐵))
9 df-rex 2901 . . . . 5 (∃𝑥𝐴 𝑤𝐵 ↔ ∃𝑥(𝑥𝐴𝑤𝐵))
10 df-rex 2901 . . . . 5 (∃𝑦𝐴 𝑤𝐵 ↔ ∃𝑦(𝑦𝐴𝑤𝐵))
118, 9, 103bitr4i 290 . . . 4 (∃𝑥𝐴 𝑤𝐵 ↔ ∃𝑦𝐴 𝑤𝐵)
1211abbii 2725 . . 3 {𝑤 ∣ ∃𝑥𝐴 𝑤𝐵} = {𝑤 ∣ ∃𝑦𝐴 𝑤𝐵}
13 df-iun 4451 . . 3 𝑥𝐴 𝐵 = {𝑤 ∣ ∃𝑥𝐴 𝑤𝐵}
14 df-iun 4451 . . 3 𝑦𝐴 𝐵 = {𝑤 ∣ ∃𝑦𝐴 𝑤𝐵}
1512, 13, 143eqtr4i 2641 . 2 𝑥𝐴 𝐵 = 𝑦𝐴 𝐵
16 bnj1143 29921 . 2 𝑦𝐴 𝐵𝐵
1715, 16eqsstri 3597 1 𝑥𝐴 𝐵𝐵
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1472  wex 1694  wcel 1976  {cab 2595  wrex 2896  wss 3539   ciun 4449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-v 3174  df-dif 3542  df-in 3546  df-ss 3553  df-nul 3874  df-iun 4451
This theorem is referenced by:  bnj1145  30121
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