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Mirrors > Home > MPE Home > Th. List > cbvrexv2 | Structured version Visualization version 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 2764 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2764 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | nfv 1843 | . 2 ⊢ Ⅎ𝑦𝜓 | |
4 | nfv 1843 | . 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 3566 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∃wrex 2913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-sbc 3436 df-csb 3534 |
This theorem is referenced by: (None) |
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