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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 2189 | . 2 ⊢ (𝑥 = 𝐴 → (𝐸 = 𝐶 ↔ 𝐸 = 𝐷)) |
3 | 2 | rspcev 2841 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐸 = 𝐷) → ∃𝑥 ∈ 𝐵 𝐸 = 𝐶) |
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: elixpsn 6728 ixpsnf1o 6729 elfir 6965 0ct 7099 ctmlemr 7100 ctssdclemn0 7102 fodju0 7138 mertenslemi1 11514 mertenslem2 11515 pcprmpw 12303 1arithlem4 12334 ctiunctlemfo 12410 elrestr 12631 restopnb 13314 mopnex 13638 metrest 13639 2sqlem2 14084 mul2sq 14085 2sqlem3 14086 2sqlem9 14093 2sqlem10 14094 |
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