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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1340 | 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 |
---|---|
bnj1340.1 | ⊢ (𝜓 → ∃𝑥𝜃) |
bnj1340.2 | ⊢ (𝜒 ↔ (𝜓 ∧ 𝜃)) |
bnj1340.3 | ⊢ (𝜓 → ∀𝑥𝜓) |
Ref | Expression |
---|---|
bnj1340 | ⊢ (𝜓 → ∃𝑥𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1340.3 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
2 | bnj1340.1 | . . 3 ⊢ (𝜓 → ∃𝑥𝜃) | |
3 | 1, 2 | bnj596 32288 | . 2 ⊢ (𝜓 → ∃𝑥(𝜓 ∧ 𝜃)) |
4 | bnj1340.2 | . 2 ⊢ (𝜒 ↔ (𝜓 ∧ 𝜃)) | |
5 | 3, 4 | bnj1198 32338 | 1 ⊢ (𝜓 → ∃𝑥𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀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 1916 ax-6 1974 ax-7 2019 ax-10 2144 ax-12 2178 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1787 df-nf 1791 |
This theorem is referenced by: bnj1450 32593 |
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