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Theorem bnj226 34209
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 3064 . 2 𝑥𝐴 𝐵𝐶
3 iunss 5048 . 2 ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)
42, 3mpbir 230 1 𝑥𝐴 𝐵𝐶
Colors of variables: wff setvar class
Syntax hints:  wral 3060  wss 3948   ciun 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-v 3475  df-in 3955  df-ss 3965  df-iun 4999
This theorem is referenced by:  bnj229  34359  bnj1128  34465  bnj1145  34468
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