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Mirrors > Home > ILE Home > Th. List > rexlimd | GIF version |
Description: Deduction from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
Ref | Expression |
---|---|
rexlimd.1 | ⊢ Ⅎ𝑥𝜑 |
rexlimd.2 | ⊢ Ⅎ𝑥𝜒 |
rexlimd.3 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
Ref | Expression |
---|---|
rexlimd | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | rexlimd.3 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) | |
3 | 1, 2 | ralrimi 2432 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
4 | rexlimd.2 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
5 | 4 | r19.23 2468 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
6 | 3, 5 | sylib 120 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Ⅎwnf 1389 ∈ wcel 1433 ∀wral 2348 ∃wrex 2349 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-ial 1467 ax-i5r 1468 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-ral 2353 df-rex 2354 |
This theorem is referenced by: rexlimdv 2476 ralxfrALT 4217 fvmptt 5283 ffnfv 5344 nneneq 6343 ac6sfi 6379 prarloclem3step 6686 prmuloc2 6757 caucvgprprlemaddq 6898 lbzbi 8701 divalglemeunn 10321 divalglemeuneg 10323 oddpwdclemdvds 10548 oddpwdclemndvds 10549 |
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