Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 19.38 | Structured version Visualization version GIF version |
Description: Theorem 19.38 of [Margaris] p. 90. The converse holds under non-freeness conditions, see 19.38a 1767 and 19.38b 1768. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2073. (Revised by Wolf Lammen, 2-Jan-2018.) |
Ref | Expression |
---|---|
19.38 | ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alnex 1706 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑) | |
2 | pm2.21 120 | . . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | |
3 | 2 | alimi 1739 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝜓)) |
4 | 1, 3 | sylbir 225 | . 2 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥(𝜑 → 𝜓)) |
5 | ala1 1741 | . 2 ⊢ (∀𝑥𝜓 → ∀𝑥(𝜑 → 𝜓)) | |
6 | 4, 5 | ja 173 | 1 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1481 ∃wex 1704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 |
This theorem depends on definitions: df-bi 197 df-ex 1705 |
This theorem is referenced by: 19.38a 1767 19.38b 1768 nfimt 1821 19.21v 1868 19.23v 1902 19.21tOLDOLD 2074 19.21tOLD 2213 bj-19.21t 32817 pm10.53 38565 |
Copyright terms: Public domain | W3C validator |