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Mirrors > Home > MPE Home > Th. List > 19.37v | Structured version Visualization version GIF version |
Description: Version of 19.37 2275 with a disjoint variable 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 1980 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | 19.3v 2085 | . . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑) | |
3 | 2 | imbi1i 341 | . 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (𝜑 → ∃𝑥𝜓)) |
4 | 1, 3 | bitri 267 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1654 ∃wex 1878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 |
This theorem depends on definitions: df-bi 199 df-ex 1879 |
This theorem is referenced by: 19.37ivOLD 2097 eqvincg 3547 axrep5 5002 fvn0ssdmfun 6604 kmlem14 9307 kmlem15 9308 bnj132 31337 bnj1098 31396 bnj150 31488 bnj865 31535 bnj996 31567 bnj1021 31576 bnj1090 31589 bnj1176 31615 bj-axrep5 33316 cnvssco 38752 refimssco 38753 19.37vv 39423 pm11.61 39432 relopabVD 39954 rmoanim 42002 |
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