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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cbvexiw | Structured version Visualization version GIF version |
Description: Change bound variable. This is to cbvexvw 2035 what cbvaliw 2004 is to cbvalvw 2034. TODO: move after cbvalivw 2005. (Contributed by BJ, 17-Mar-2020.) |
Ref | Expression |
---|---|
bj-cbvexiw.1 | ⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) |
bj-cbvexiw.2 | ⊢ (𝜑 → ∀𝑦𝜑) |
bj-cbvexiw.3 | ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
bj-cbvexiw | ⊢ (∃𝑥𝜑 → ∃𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-cbvexiw.1 | . 2 ⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) | |
2 | bj-cbvexiw.2 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) | |
3 | bj-cbvexiw.3 | . . 3 ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓)) | |
4 | 2, 3 | spimew 1965 | . 2 ⊢ (𝜑 → ∃𝑦𝜓) |
5 | 1, 4 | bj-sylge 33854 | 1 ⊢ (∃𝑥𝜑 → ∃𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 ∃wex 1771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-6 1961 |
This theorem depends on definitions: df-bi 208 df-ex 1772 |
This theorem is referenced by: bj-cbvexivw 33902 bj-cbvexw 33906 |
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