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Mirrors > Home > MPE Home > Th. List > cbvmovw | Structured version Visualization version GIF version |
Description: Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvmo 2605 and cbvmow 2603 for versions with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 9-Mar-1995.) (Revised by Gino Giotto, 30-Sep-2024.) |
Ref | Expression |
---|---|
cbvmovw.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvmovw | ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvmovw.1 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | equequ1 2028 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) | |
3 | 1, 2 | imbi12d 345 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜓 → 𝑦 = 𝑧))) |
4 | 3 | cbvalvw 2039 | . . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑦(𝜓 → 𝑦 = 𝑧)) |
5 | 4 | exbii 1850 | . 2 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) |
6 | df-mo 2540 | . 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧)) | |
7 | df-mo 2540 | . 2 ⊢ (∃*𝑦𝜓 ↔ ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) | |
8 | 5, 6, 7 | 3bitr4i 303 | 1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1537 ∃wex 1782 ∃*wmo 2538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 df-mo 2540 |
This theorem is referenced by: cbveuvw 2606 cbvrmovw 3385 |
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