Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 19.36 | Structured version Visualization version GIF version |
Description: Theorem 19.36 of [Margaris] p. 90. See 19.36v 1985 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.) |
Ref | Expression |
---|---|
19.36.1 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
19.36 | ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.35 1869 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | 19.36.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | 2 | 19.9 2195 | . . 3 ⊢ (∃𝑥𝜓 ↔ 𝜓) |
4 | 3 | imbi2i 337 | . 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
5 | 1, 4 | bitri 276 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∀wal 1526 ∃wex 1771 Ⅎwnf 1775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-ex 1772 df-nf 1776 |
This theorem is referenced by: 19.36i 2223 19.12vv 2359 spcimgft 3583 19.12b 32943 |
Copyright terms: Public domain | W3C validator |