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Mirrors > Home > MPE Home > Th. List > cbvrexv | Structured version Visualization version GIF version |
Description: Change the bound variable of a restricted existential quantifier using implicit substitution. See cbvrexvw 3450 based on fewer axioms , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker cbvrexvw 3450 when possible. (Contributed by NM, 2-Jun-1998.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvralv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvrexv | ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1911 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1911 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvralv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvrex 3446 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∃wrex 3139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-10 2141 ax-11 2157 ax-12 2173 ax-13 2386 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 |
This theorem is referenced by: cbvrex2v 3465 cygablOLD 19005 rexlimdvaacbv 40551 |
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