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Mirrors > Home > ILE Home > Th. List > rspceeqv | GIF version |
Description: Restricted existential specialization in an equality, using implicit substitution. (Contributed by BJ, 2-Sep-2022.) |
Ref | Expression |
---|---|
rspceeqv.1 | ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
rspceeqv | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐸 = 𝐷) → ∃𝑥 ∈ 𝐵 𝐸 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspceeqv.1 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷) | |
2 | 1 | eqeq2d 2151 | . 2 ⊢ (𝑥 = 𝐴 → (𝐸 = 𝐶 ↔ 𝐸 = 𝐷)) |
3 | 2 | rspcev 2789 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐸 = 𝐷) → ∃𝑥 ∈ 𝐵 𝐸 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 |
This theorem is referenced by: elixpsn 6629 ixpsnf1o 6630 elfir 6861 0ct 6992 ctmlemr 6993 ctssdclemn0 6995 fodju0 7019 mertenslemi1 11304 mertenslem2 11305 ctiunctlemfo 11952 elrestr 12128 restopnb 12350 mopnex 12674 metrest 12675 |
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