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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 2385. (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 34652 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ∃𝑥(𝜑 → 𝜓)) | |
2 | 19.35 1885 | . . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
3 | 1, 2 | sylib 221 | . 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 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-5 1918 ax-6 1976 ax-7 2018 ax-12 2177 ax-13 2371 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 |
This theorem is referenced by: bj-cbv3ta 34654 |
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