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Mirrors > Home > MPE Home > Th. List > 19.37v | Structured version Visualization version GIF version |
Description: Version of 19.37 2234 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 1878 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | 19.3v 1986 | . . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑) | |
3 | 2 | imbi1i 352 | . 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (𝜑 → ∃𝑥𝜓)) |
4 | 1, 3 | bitri 277 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 |
This theorem depends on definitions: df-bi 209 df-ex 1781 |
This theorem is referenced by: spc3egv 3606 eqvincg 3643 rmoanim 3880 rmoanimALT 3881 axrep5 5198 fvn0ssdmfun 6844 kmlem14 9591 kmlem15 9592 bnj132 31998 bnj1098 32057 bnj150 32150 bnj865 32197 bnj996 32230 bnj1021 32240 bnj1090 32253 bnj1176 32279 sn-axrep5v 39115 cnvssco 39973 refimssco 39974 19.37vv 40724 pm11.61 40732 relopabVD 41242 |
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