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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 32726 | . 2 ⊢ (𝜓 → ∃𝑥(𝜓 ∧ 𝜃)) |
4 | bnj1340.2 | . 2 ⊢ (𝜒 ↔ (𝜓 ∧ 𝜃)) | |
5 | 3, 4 | bnj1198 32775 | 1 ⊢ (𝜓 → ∃𝑥𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1537 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 df-nf 1787 |
This theorem is referenced by: bnj1450 33030 |
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