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Mirrors > Home > MPE Home > Th. List > 19.37v | Structured version Visualization version GIF version |
Description: Version of 19.37 2087 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 1794 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | 19.3v 1884 | . . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑) | |
3 | 2 | imbi1i 338 | . 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (𝜑 → ∃𝑥𝜓)) |
4 | 1, 3 | bitri 263 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 |
This theorem depends on definitions: df-bi 196 df-ex 1696 |
This theorem is referenced by: 19.37iv 1898 axrep5 4704 fvn0ssdmfun 6258 kmlem14 8868 kmlem15 8869 eqvincg 28698 bnj132 30046 bnj1098 30108 bnj150 30200 bnj865 30247 bnj996 30279 bnj1021 30288 bnj1090 30301 bnj1176 30327 bj-axrep5 31980 cnvssco 36931 refimssco 36932 19.37vv 37606 pm11.61 37615 relopabVD 38159 rmoanim 39828 |
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