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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1209 | 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 |
---|---|
bnj1209.1 | ⊢ (𝜒 → ∃𝑥 ∈ 𝐵 𝜑) |
bnj1209.2 | ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑)) |
Ref | Expression |
---|---|
bnj1209 | ⊢ (𝜒 → ∃𝑥𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1209.1 | . . . . 5 ⊢ (𝜒 → ∃𝑥 ∈ 𝐵 𝜑) | |
2 | 1 | bnj1196 30119 | . . . 4 ⊢ (𝜒 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) |
3 | 2 | ancli 572 | . . 3 ⊢ (𝜒 → (𝜒 ∧ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))) |
4 | 19.42v 1905 | . . 3 ⊢ (∃𝑥(𝜒 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑)) ↔ (𝜒 ∧ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))) | |
5 | 3, 4 | sylibr 223 | . 2 ⊢ (𝜒 → ∃𝑥(𝜒 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑))) |
6 | bnj1209.2 | . . 3 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑)) | |
7 | 3anass 1035 | . . 3 ⊢ ((𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝜒 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑))) | |
8 | 6, 7 | bitri 263 | . 2 ⊢ (𝜃 ↔ (𝜒 ∧ (𝑥 ∈ 𝐵 ∧ 𝜑))) |
9 | 5, 8 | bnj1198 30120 | 1 ⊢ (𝜒 → ∃𝑥𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∃wex 1695 ∈ wcel 1977 ∃wrex 2897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 |
This theorem depends on definitions: df-bi 196 df-an 385 df-3an 1033 df-ex 1696 df-rex 2902 |
This theorem is referenced by: bnj1501 30389 bnj1523 30393 |
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