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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 2390. Use the weaker cbvmow 2688 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 2687 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑) |
3 | cbvmo.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | cbvmo.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | sbie 2544 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
6 | 5 | mobii 2631 | . 2 ⊢ (∃*𝑦[𝑦 / 𝑥]𝜑 ↔ ∃*𝑦𝜓) |
7 | 2, 6 | bitri 277 | 1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 Ⅎwnf 1784 [wsb 2069 ∃*wmo 2620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 |
This theorem is referenced by: cbveuALT 2692 |
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