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Theorem bnj226 32112
 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 3121 . 2 𝑥𝐴 𝐵𝐶
3 iunss 4935 . 2 ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)
42, 3mpbir 234 1 𝑥𝐴 𝐵𝐶
 Colors of variables: wff setvar class Syntax hints:  ∀wral 3109   ⊆ wss 3884  ∪ ciun 4884 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-in 3891  df-ss 3901  df-iun 4886 This theorem is referenced by:  bnj229  32264  bnj1128  32370  bnj1145  32373
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