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Mirrors > Home > MPE Home > Th. List > cbvmo | 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 2379. Use the weaker cbvmow 2663 when possible. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Wolf Lammen, 4-Jan-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvmo.1 | ⊢ Ⅎ𝑦𝜑 |
cbvmo.2 | ⊢ Ⅎ𝑥𝜓 |
cbvmo.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvmo | ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvmo.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | sb8mo 2662 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑) |
3 | cbvmo.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | cbvmo.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | sbie 2521 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
6 | 5 | mobii 2606 | . 2 ⊢ (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ ∃*𝑦𝜓) |
7 | 2, 6 | bitri 278 | 1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 Ⅎwnf 1785 [wsb 2069 ∃*wmo 2596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-11 2158 ax-12 2175 ax-13 2379 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 |
This theorem is referenced by: cbveuALT 2669 |
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