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 ( 𝜑 → 𝜓 ) | |
bnj1023.2 | ⊢ ( 𝜓 → 𝜒 ) | ||
Assertion | bnj1023 | ⊢ ∃ 𝑥 ( 𝜑 → 𝜒 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1023.1 | ⊢ ∃ 𝑥 ( 𝜑 → 𝜓 ) | |
2 | bnj1023.2 | ⊢ ( 𝜓 → 𝜒 ) | |
3 | 2 | a1i | ⊢ ( ( 𝜑 → 𝜓 ) → ( 𝜓 → 𝜒 ) ) |
4 | 3 | ax-gen | ⊢ ∀ 𝑥 ( ( 𝜑 → 𝜓 ) → ( 𝜓 → 𝜒 ) ) |
5 | exintr | ⊢ ( ∀ 𝑥 ( ( 𝜑 → 𝜓 ) → ( 𝜓 → 𝜒 ) ) → ( ∃ 𝑥 ( 𝜑 → 𝜓 ) → ∃ 𝑥 ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜒 ) ) ) ) | |
6 | 4 1 5 | mp2 | ⊢ ∃ 𝑥 ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜒 ) ) |
7 | pm3.33 | ⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜓 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) | |
8 | 6 7 | bnj101 | ⊢ ∃ 𝑥 ( 𝜑 → 𝜒 ) |