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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-exlimmp | GIF version |
Description: Lemma for bj-vtoclgf 13770. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-exlimmp.nf | ⊢ Ⅎ𝑥𝜓 |
bj-exlimmp.min | ⊢ (𝜒 → 𝜑) |
Ref | Expression |
---|---|
bj-exlimmp | ⊢ (∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜒 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 1534 | . 2 ⊢ Ⅎ𝑥∀𝑥(𝜒 → (𝜑 → 𝜓)) | |
2 | bj-exlimmp.nf | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | bj-exlimmp.min | . . . . 5 ⊢ (𝜒 → 𝜑) | |
4 | idd 21 | . . . . 5 ⊢ (𝜒 → (𝜓 → 𝜓)) | |
5 | 3, 4 | embantd 56 | . . . 4 ⊢ (𝜒 → ((𝜑 → 𝜓) → 𝜓)) |
6 | 5 | a2i 11 | . . 3 ⊢ ((𝜒 → (𝜑 → 𝜓)) → (𝜒 → 𝜓)) |
7 | 6 | sps 1530 | . 2 ⊢ (∀𝑥(𝜒 → (𝜑 → 𝜓)) → (𝜒 → 𝜓)) |
8 | 1, 2, 7 | exlimd 1590 | 1 ⊢ (∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜒 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1346 Ⅎwnf 1453 ∃wex 1485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-gen 1442 ax-ie2 1487 ax-4 1503 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 df-nf 1454 |
This theorem is referenced by: bj-vtoclgft 13769 |
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