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Mirrors > Home > MPE Home > Th. List > rspcedvdw | Structured version Visualization version GIF version |
Description: Version of rspcedvd 3608 where the implicit substitution hypothesis does not have an antecedent, which also avoids a disjoint variable condition on 𝜑, 𝑥. (Contributed by SN, 20-Aug-2024.) |
Ref | Expression |
---|---|
rspcedvdw.s | ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) |
rspcedvdw.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
rspcedvdw.2 | ⊢ (𝜑 → 𝜒) |
Ref | Expression |
---|---|
rspcedvdw | ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcedvdw.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
2 | rspcedvdw.2 | . 2 ⊢ (𝜑 → 𝜒) | |
3 | rspcedvdw.s | . . 3 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) | |
4 | 3 | rspcev 3606 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜒) → ∃𝑥 ∈ 𝐵 𝜓) |
5 | 1, 2, 4 | syl2anc 583 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 |
This theorem is referenced by: opprring 20249 pzriprnglem7 21374 pzriprnglem13 21380 pzriprnglem14 21381 pzriprngALT 21382 remulscllem1 28183 remulscllem2 28184 ply1degltdimlem 33225 irredminply 33293 algextdeglem4 33297 algextdeglem8 33301 flt4lem2 41967 flt4lem7 41979 |
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