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Mirrors > Home > MPE Home > Th. List > cbvrex | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2363. Use the weaker cbvrexw 3296 when possible. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvral.1 | ⊢ Ⅎ𝑦𝜑 |
cbvral.2 | ⊢ Ⅎ𝑥𝜓 |
cbvral.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvrex | ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2895 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2895 | . 2 ⊢ Ⅎ𝑦𝐴 | |
3 | cbvral.1 | . 2 ⊢ Ⅎ𝑦𝜑 | |
4 | cbvral.2 | . 2 ⊢ Ⅎ𝑥𝜓 | |
5 | cbvral.3 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 1, 2, 3, 4, 5 | cbvrexf 3349 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 Ⅎwnf 1777 ∃wrex 3062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-10 2129 ax-11 2146 ax-12 2163 ax-13 2363 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 |
This theorem is referenced by: cbvrexv 3353 cbvrexsv 3355 cbvrmo 3417 cbviung 5032 |
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