![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 19.37v | Structured version Visualization version GIF version |
Description: Version of 19.37 2138 with a dv condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
19.37v | ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.35 1845 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | 19.3v 1954 | . . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑) | |
3 | 2 | imbi1i 338 | . 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (𝜑 → ∃𝑥𝜓)) |
4 | 1, 3 | bitri 264 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1521 ∃wex 1744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 |
This theorem depends on definitions: df-bi 197 df-ex 1745 |
This theorem is referenced by: 19.37ivOLD 1967 eqvincg 3360 axrep5 4809 fvn0ssdmfun 6390 kmlem14 9023 kmlem15 9024 bnj132 30920 bnj1098 30980 bnj150 31072 bnj865 31119 bnj996 31151 bnj1021 31160 bnj1090 31173 bnj1176 31199 bj-axrep5 32917 cnvssco 38229 refimssco 38230 19.37vv 38901 pm11.61 38910 relopabVD 39451 rmoanim 41500 |
Copyright terms: Public domain | W3C validator |