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Theorem bnj1023 29898
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1023.1 𝑥(𝜑𝜓)
bnj1023.2 (𝜓𝜒)
Assertion
Ref Expression
bnj1023 𝑥(𝜑𝜒)

Proof of Theorem bnj1023
StepHypRef Expression
1 bnj1023.2 . . . . 5 (𝜓𝜒)
21a1i 11 . . . 4 ((𝜑𝜓) → (𝜓𝜒))
32ax-gen 1712 . . 3 𝑥((𝜑𝜓) → (𝜓𝜒))
4 bnj1023.1 . . 3 𝑥(𝜑𝜓)
5 exintr 1809 . . 3 (∀𝑥((𝜑𝜓) → (𝜓𝜒)) → (∃𝑥(𝜑𝜓) → ∃𝑥((𝜑𝜓) ∧ (𝜓𝜒))))
63, 4, 5mp2 9 . 2 𝑥((𝜑𝜓) ∧ (𝜓𝜒))
7 pm3.33 606 . 2 (((𝜑𝜓) ∧ (𝜓𝜒)) → (𝜑𝜒))
86, 7bnj101 29836 1 𝑥(𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by:  bnj1098  29901  bnj1110  30097  bnj1118  30099  bnj1128  30105  bnj1145  30108
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