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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1238 | 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 |
---|---|
bnj1238.1 | ⊢ (𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒)) |
Ref | Expression |
---|---|
bnj1238 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1238.1 | . 2 ⊢ (𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒)) | |
2 | bnj1239 32081 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 𝜓) | |
3 | 1, 2 | sylbi 219 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∃wrex 3142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-ral 3146 df-rex 3147 |
This theorem is referenced by: bnj1245 32290 bnj1256 32291 bnj1259 32292 bnj1311 32300 bnj1371 32305 |
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