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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1521 | 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 |
---|---|
bnj1521.1 | ⊢ (𝜒 → ∃𝑥 ∈ 𝐵 𝜑) |
bnj1521.2 | ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑)) |
bnj1521.3 | ⊢ (𝜒 → ∀𝑥𝜒) |
Ref | Expression |
---|---|
bnj1521 | ⊢ (𝜒 → ∃𝑥𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1521.1 | . . 3 ⊢ (𝜒 → ∃𝑥 ∈ 𝐵 𝜑) | |
2 | 1 | bnj1196 32068 | . 2 ⊢ (𝜒 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) |
3 | bnj1521.2 | . 2 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑)) | |
4 | bnj1521.3 | . 2 ⊢ (𝜒 → ∀𝑥𝜒) | |
5 | 2, 3, 4 | bnj1345 32098 | 1 ⊢ (𝜒 → ∃𝑥𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 ∀wal 1535 ∃wex 1780 ∈ wcel 2114 ∃wrex 3141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-3an 1085 df-ex 1781 df-nf 1785 df-rex 3146 |
This theorem is referenced by: bnj1204 32286 bnj1311 32298 bnj1398 32308 bnj1408 32310 bnj1450 32324 bnj1312 32332 bnj1501 32341 bnj1523 32345 |
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