Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-ex | GIF version |
Description: Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1591 and 19.9ht 1634 or 19.23ht 1490). (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ex | ⊢ (∃𝑥𝜑 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-ie2 1487 | . . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜑) ↔ (∃𝑥𝜑 → 𝜑))) | |
2 | ax-17 1519 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | 1, 2 | mpg 1444 | . 2 ⊢ (∀𝑥(𝜑 → 𝜑) ↔ (∃𝑥𝜑 → 𝜑)) |
4 | id 19 | . 2 ⊢ (𝜑 → 𝜑) | |
5 | 3, 4 | mpgbi 1445 | 1 ⊢ (∃𝑥𝜑 → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1346 ∃wex 1485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-gen 1442 ax-ie2 1487 ax-17 1519 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: bj-d0clsepcl 13922 bj-inf2vnlem1 13967 bj-nn0sucALT 13975 |
Copyright terms: Public domain | W3C validator |