Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rr-spce | Structured version Visualization version GIF version |
Description: Prove an existential. (Contributed by Rohan Ridenour, 12-Aug-2023.) |
Ref | Expression |
---|---|
rr-spce.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓) |
rr-spce.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
rr-spce | ⊢ (𝜑 → ∃𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rr-spce.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | 1 | elexd 3449 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
3 | isset 3442 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
4 | 2, 3 | sylib 217 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴) |
5 | rr-spce.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓) | |
6 | 5 | ex 413 | . . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → 𝜓)) |
7 | 6 | eximdv 1920 | . 2 ⊢ (𝜑 → (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜓)) |
8 | 4, 7 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 Vcvv 3429 |
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-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3431 |
This theorem is referenced by: grumnudlem 41884 |
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