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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-spimt2 | Structured version Visualization version GIF version | ||
| Description: A step in the proof of spimt 2391. (Contributed by BJ, 2-May-2019.) |
| Ref | Expression |
|---|---|
| bj-spimt2 | ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → 𝜓))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-alequex 36785 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ∃𝑥(𝜑 → 𝜓)) | |
| 2 | 19.35 1877 | . . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
| 3 | 1, 2 | sylib 218 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∃𝑥𝜓)) |
| 4 | 3 | imim1d 82 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → 𝜓))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 ∃wex 1779 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-12 2177 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 |
| This theorem is referenced by: bj-cbv3ta 36787 |
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