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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 484 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜑) | |
4 | 2, 3 | 2thd 265 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜑)) |
5 | id 22 | . 2 ⊢ (𝜑 → 𝜑) | |
6 | 1, 4, 5 | rspcedvd 3585 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3070 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 |
This theorem is referenced by: elrnmptdv 5921 aks4d1p8d2 40592 mnuprdlem3 42646 mnurndlem1 42653 grumnudlem 42657 grumnud 42658 inaex 42669 gruex 42670 |
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