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Mirrors > Home > MPE Home > Th. List > rspcime | Structured version Visualization version GIF version |
Description: Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
rspcime.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓) |
rspcime.2 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
Ref | Expression |
---|---|
rspcime | ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcime.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
2 | rspcime.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓) | |
3 | simpl 486 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜑) | |
4 | 2, 3 | 2thd 268 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜑)) |
5 | id 22 | . 2 ⊢ (𝜑 → 𝜑) | |
6 | 1, 4, 5 | rspcedvd 3530 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 |
This theorem is referenced by: elrnmptdv 5816 mnuprdlem3 41506 mnurndlem1 41513 grumnudlem 41517 grumnud 41518 inaex 41529 gruex 41530 |
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