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Mirrors > Home > MPE Home > Th. List > cbvmow | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. Version of cbvmo 2607 with a disjoint variable condition, which does not require ax-10 2141, ax-13 2380. (Contributed by NM, 9-Mar-1995.) (Revised by GG, 23-May-2024.) |
Ref | Expression |
---|---|
cbvmow.1 | ⊢ Ⅎ𝑦𝜑 |
cbvmow.2 | ⊢ Ⅎ𝑥𝜓 |
cbvmow.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvmow | ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvmow.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1913 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 = 𝑧 | |
3 | 1, 2 | nfim 1895 | . . . 4 ⊢ Ⅎ𝑦(𝜑 → 𝑥 = 𝑧) |
4 | cbvmow.2 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
5 | nfv 1913 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 = 𝑧 | |
6 | 4, 5 | nfim 1895 | . . . 4 ⊢ Ⅎ𝑥(𝜓 → 𝑦 = 𝑧) |
7 | cbvmow.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
8 | equequ1 2024 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) | |
9 | 7, 8 | imbi12d 344 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜓 → 𝑦 = 𝑧))) |
10 | 3, 6, 9 | cbvalv1 2347 | . . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑦(𝜓 → 𝑦 = 𝑧)) |
11 | 10 | exbii 1846 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) |
12 | df-mo 2543 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)) | |
13 | df-mo 2543 | . 2 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) | |
14 | 11, 12, 13 | 3bitr4i 303 | 1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 ∃wex 1777 Ⅎwnf 1781 ∃*wmo 2541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-11 2158 ax-12 2178 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1778 df-nf 1782 df-mo 2543 |
This theorem is referenced by: cbveuw 2609 cbvrmow 3417 dffun6f 6591 opabiotafun 7002 2ndcdisj 23485 cbvdisjf 32593 phpreu 37564 mo0sn 48547 isthincd2lem1 48694 |
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