Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3623 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-cleq 2813 df-clel 2892 df-ral 3142 df-rex 3143 |
This theorem is referenced by: elrnmptdv 5826 mnuprdlem3 40684 mnurndlem1 40691 grumnudlem 40695 grumnud 40696 inaex 40707 gruex 40708 |
Copyright terms: Public domain | W3C validator |