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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sblem2 | Structured version Visualization version GIF version |
Description: Lemma for substitution. (Contributed by BJ, 23-Jul-2023.) |
Ref | Expression |
---|---|
bj-sblem2 | ⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) → ((∃𝑥𝜑 → 𝜒) → ∀𝑥(𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.23v 1948 | . 2 ⊢ (∀𝑥(𝜑 → 𝜒) ↔ (∃𝑥𝜑 → 𝜒)) | |
2 | ax-2 7 | . . 3 ⊢ ((𝜑 → (𝜒 → 𝜓)) → ((𝜑 → 𝜒) → (𝜑 → 𝜓))) | |
3 | 2 | al2imi 1822 | . 2 ⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) → (∀𝑥(𝜑 → 𝜒) → ∀𝑥(𝜑 → 𝜓))) |
4 | 1, 3 | syl5bir 246 | 1 ⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) → ((∃𝑥𝜑 → 𝜒) → ∀𝑥(𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 ∃wex 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 |
This theorem depends on definitions: df-bi 210 df-ex 1787 |
This theorem is referenced by: bj-sbievw2 34650 |
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