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Mirrors > Home > ILE Home > Th. List > rspcedeq2vd | GIF version |
Description: Restricted existential specialization, using implicit substitution. Variant of rspcedvd 2847 for equations, in which the right hand side depends on the quantified variable. (Contributed by AV, 24-Dec-2019.) |
Ref | Expression |
---|---|
rspcedeqvd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
rspcedeqvd.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
rspcedeq2vd | ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcedeqvd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
2 | rspcedeqvd.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷) | |
3 | 2 | eqcomd 2183 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐶) |
4 | 3 | eqeq2d 2189 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐶 = 𝐷 ↔ 𝐶 = 𝐶)) |
5 | eqidd 2178 | . 2 ⊢ (𝜑 → 𝐶 = 𝐶) | |
6 | 1, 4, 5 | rspcedvd 2847 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝐶 = 𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2739 |
This theorem is referenced by: (None) |
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