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Theorem bnj226 34748
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 3065 . 2 𝑥𝐴 𝐵𝐶
3 iunss 5045 . 2 ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)
42, 3mpbir 231 1 𝑥𝐴 𝐵𝐶
Colors of variables: wff setvar class
Syntax hints:  wral 3061  wss 3951   ciun 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-ss 3968  df-iun 4993
This theorem is referenced by:  bnj229  34898  bnj1128  35004  bnj1145  35007
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