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Mirrors > Home > MPE Home > Th. List > exintr | Structured version Visualization version GIF version |
Description: Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.) (Proof shortened by BJ, 16-Sep-2022.) |
Ref | Expression |
---|---|
exintr | ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancl 548 | . 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜑 ∧ 𝜓))) | |
2 | 1 | aleximi 1833 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1536 ∃wex 1781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 |
This theorem is referenced by: equs4v 2006 equs4 2427 eupickbi 2698 barbarilem 2730 ceqsex 3488 r19.2z 4398 pwpw0 4706 pwsnOLD 4793 bnj1023 32162 bnj1109 32168 pm10.55 41073 |
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