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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1397 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1397.1 | ⊢ (𝜑 → ∃𝑥𝜓) |
bnj1397.2 | ⊢ (𝜓 → ∀𝑥𝜓) |
Ref | Expression |
---|---|
bnj1397 | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1397.1 | . 2 ⊢ (𝜑 → ∃𝑥𝜓) | |
2 | bnj1397.2 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
3 | 2 | 19.9h 2290 | . 2 ⊢ (∃𝑥𝜓 ↔ 𝜓) |
4 | 1, 3 | sylib 221 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 ∃wex 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-10 2145 ax-12 2179 |
This theorem depends on definitions: df-bi 210 df-ex 1787 df-nf 1791 |
This theorem is referenced by: bnj1398 32585 bnj1408 32587 bnj1450 32601 bnj1501 32618 |
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