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Mirrors > Home > ILE Home > Th. List > cbvrexv2 | GIF version |
Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
cbvralv2.1 | ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) |
cbvralv2.2 | ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
cbvrexv2 | ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2235 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2235 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | nfv 1473 | . 2 ⊢ Ⅎ𝑦𝜓 | |
4 | nfv 1473 | . 2 ⊢ Ⅎ𝑥𝜒 | |
5 | cbvralv2.2 | . 2 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
6 | cbvralv2.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | |
7 | 1, 2, 3, 4, 5, 6 | cbvrexcsf 3005 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1296 ∃wrex 2371 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-rex 2376 df-sbc 2855 df-csb 2948 |
This theorem is referenced by: (None) |
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