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Mirrors > Home > ILE Home > Th. List > ceqsexg | GIF version |
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.) |
Ref | Expression |
---|---|
ceqsexg.1 | ⊢ Ⅎ𝑥𝜓 |
ceqsexg.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ceqsexg | ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2282 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfe1 1473 | . . 3 ⊢ Ⅎ𝑥∃𝑥(𝑥 = 𝐴 ∧ 𝜑) | |
3 | ceqsexg.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | 2, 3 | nfbi 1569 | . 2 ⊢ Ⅎ𝑥(∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) |
5 | ceqex 2816 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))) | |
6 | ceqsexg.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | 5, 6 | bibi12d 234 | . 2 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜑) ↔ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓))) |
8 | biid 170 | . 2 ⊢ (𝜑 ↔ 𝜑) | |
9 | 1, 4, 7, 8 | vtoclgf 2747 | 1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 Ⅎwnf 1437 ∃wex 1469 ∈ wcel 1481 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 |
This theorem is referenced by: ceqsexgv 2818 |
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