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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 485 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜑) | |
4 | 2, 3 | 2thd 267 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜑)) |
5 | id 22 | . 2 ⊢ (𝜑 → 𝜑) | |
6 | 1, 4, 5 | rspcedvd 3626 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 |
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-8 2116 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-cleq 2814 df-clel 2893 df-ral 3143 df-rex 3144 |
This theorem is referenced by: elrnmptdv 5833 mnuprdlem3 40659 mnurndlem1 40666 grumnudlem 40670 grumnud 40671 inaex 40682 gruex 40683 |
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