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Theorem bnj1521 30175
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1521.1 (𝜒 → ∃𝑥𝐵 𝜑)
bnj1521.2 (𝜃 ↔ (𝜒𝑥𝐵𝜑))
bnj1521.3 (𝜒 → ∀𝑥𝜒)
Assertion
Ref Expression
bnj1521 (𝜒 → ∃𝑥𝜃)

Proof of Theorem bnj1521
StepHypRef Expression
1 bnj1521.1 . . 3 (𝜒 → ∃𝑥𝐵 𝜑)
21bnj1196 30119 . 2 (𝜒 → ∃𝑥(𝑥𝐵𝜑))
3 bnj1521.2 . 2 (𝜃 ↔ (𝜒𝑥𝐵𝜑))
4 bnj1521.3 . 2 (𝜒 → ∀𝑥𝜒)
52, 3, 4bnj1345 30149 1 (𝜒 → ∃𝑥𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031  ∀wal 1473  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-3an 1033  df-ex 1696  df-nf 1701  df-rex 2902 This theorem is referenced by:  bnj1204  30334  bnj1311  30346  bnj1398  30356  bnj1408  30358  bnj1450  30372  bnj1312  30380  bnj1501  30389  bnj1523  30393
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