Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cbvexiw | Structured version Visualization version GIF version |
Description: Change bound variable. This is to cbvexvw 2045 what cbvaliw 2014 is to cbvalvw 2044. TODO: move after cbvalivw 2015. (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 1980 | . 2 ⊢ (𝜑 → ∃𝑦𝜓) |
5 | 1, 4 | bj-sylge 34542 | 1 ⊢ (∃𝑥𝜑 → ∃𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 ∃wex 1787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-6 1976 |
This theorem depends on definitions: df-bi 210 df-ex 1788 |
This theorem is referenced by: bj-cbvexivw 34590 bj-cbvexw 34594 |
Copyright terms: Public domain | W3C validator |