Metamath Proof Explorer


Theorem bnj1023

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1023.1 𝑥 ( 𝜑𝜓 )
bnj1023.2 ( 𝜓𝜒 )
Assertion bnj1023 𝑥 ( 𝜑𝜒 )

Proof

Step Hyp Ref Expression
1 bnj1023.1 𝑥 ( 𝜑𝜓 )
2 bnj1023.2 ( 𝜓𝜒 )
3 2 a1i ( ( 𝜑𝜓 ) → ( 𝜓𝜒 ) )
4 3 ax-gen 𝑥 ( ( 𝜑𝜓 ) → ( 𝜓𝜒 ) )
5 exintr ( ∀ 𝑥 ( ( 𝜑𝜓 ) → ( 𝜓𝜒 ) ) → ( ∃ 𝑥 ( 𝜑𝜓 ) → ∃ 𝑥 ( ( 𝜑𝜓 ) ∧ ( 𝜓𝜒 ) ) ) )
6 4 1 5 mp2 𝑥 ( ( 𝜑𝜓 ) ∧ ( 𝜓𝜒 ) )
7 pm3.33 ( ( ( 𝜑𝜓 ) ∧ ( 𝜓𝜒 ) ) → ( 𝜑𝜒 ) )
8 6 7 bnj101 𝑥 ( 𝜑𝜒 )