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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 1560 and 19.9ht 1603 or 19.23ht 1456). (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ex | ⊢ (∃𝑥𝜑 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-ie2 1453 | . . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜑) ↔ (∃𝑥𝜑 → 𝜑))) | |
2 | ax-17 1489 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | 1, 2 | mpg 1410 | . 2 ⊢ (∀𝑥(𝜑 → 𝜑) ↔ (∃𝑥𝜑 → 𝜑)) |
4 | id 19 | . 2 ⊢ (𝜑 → 𝜑) | |
5 | 3, 4 | mpgbi 1411 | 1 ⊢ (∃𝑥𝜑 → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1312 ∃wex 1451 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-gen 1408 ax-ie2 1453 ax-17 1489 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: bj-d0clsepcl 12815 bj-inf2vnlem1 12860 bj-nn0sucALT 12868 |
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