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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sblem1 | Structured version Visualization version GIF version |
Description: Lemma for substitution. (Contributed by BJ, 23-Jul-2023.) |
Ref | Expression |
---|---|
bj-sblem1 | ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-2 7 | . . 3 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | |
2 | 1 | al2imi 1822 | . 2 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜒))) |
3 | 19.23v 1949 | . 2 ⊢ (∀𝑥(𝜑 → 𝜒) ↔ (∃𝑥𝜑 → 𝜒)) | |
4 | 2, 3 | syl6ib 254 | 1 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜒))) |
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 1917 |
This theorem depends on definitions: df-bi 210 df-ex 1787 |
This theorem is referenced by: bj-sbievw1 34660 |
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