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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 3359 based on fewer axioms , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvrexvw 3359 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 1922 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1922 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvralv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvrex 3355 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∃wrex 3062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-10 2141 ax-11 2158 ax-12 2175 ax-13 2371 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2071 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 |
This theorem is referenced by: cbvrex2v 3375 cygablOLD 19276 rexlimdvaacbv 41494 |
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