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Mirrors > Home > ILE Home > Th. List > 19.23h | GIF version |
Description: Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.) |
Ref | Expression |
---|---|
19.23h.1 | ⊢ (𝜓 → ∀𝑥𝜓) |
Ref | Expression |
---|---|
19.23h | ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.23h.1 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
2 | 1 | ax-gen 1406 | . 2 ⊢ ∀𝑥(𝜓 → ∀𝑥𝜓) |
3 | 19.23ht 1454 | . 2 ⊢ (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) | |
4 | 2, 3 | ax-mp 7 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1310 ∃wex 1449 |
This theorem was proved from axioms: ax-mp 7 ax-gen 1406 ax-ie2 1451 |
This theorem is referenced by: alnex 1456 19.8a 1550 exlimih 1553 exlimdh 1556 nf2 1627 equs5or 1782 19.23v 1835 pm11.53 1847 |
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