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Theorem bnj226 33682
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj226.1 𝐵𝐶
Assertion
Ref Expression
bnj226 𝑥𝐴 𝐵𝐶
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem bnj226
StepHypRef Expression
1 bnj226.1 . . 3 𝐵𝐶
21rgenw 3066 . 2 𝑥𝐴 𝐵𝐶
3 iunss 5046 . 2 ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)
42, 3mpbir 230 1 𝑥𝐴 𝐵𝐶
Colors of variables: wff setvar class
Syntax hints:  wral 3062  wss 3946   ciun 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-v 3477  df-in 3953  df-ss 3963  df-iun 4997
This theorem is referenced by:  bnj229  33832  bnj1128  33938  bnj1145  33941
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