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Mirrors > Home > ILE Home > Th. List > r19.2m | GIF version |
Description: Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1631). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.) (Revised by Jim Kingdon, 7-Apr-2023.) |
Ref | Expression |
---|---|
r19.2m | ⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2231 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
2 | 1 | cbvexv 1911 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑧 𝑧 ∈ 𝐴) |
3 | eleq1w 2231 | . . . 4 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
4 | 3 | cbvexv 1911 | . . 3 ⊢ (∃𝑧 𝑧 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴) |
5 | 2, 4 | bitri 183 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴) |
6 | df-ral 2453 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
7 | exintr 1627 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) | |
8 | 6, 7 | sylbi 120 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) |
9 | df-rex 2454 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
10 | 8, 9 | syl6ibr 161 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 𝜑)) |
11 | 10 | impcom 124 | . 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑) |
12 | 5, 11 | sylanbr 283 | 1 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1346 ∃wex 1485 ∈ wcel 2141 ∀wral 2448 ∃wrex 2449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 df-clel 2166 df-ral 2453 df-rex 2454 |
This theorem is referenced by: intssunim 3851 riinm 3943 iinexgm 4138 xpiindim 4746 cnviinm 5150 eusvobj2 5836 iinerm 6581 suplocexprlemml 7665 rexfiuz 10940 r19.2uz 10944 climuni 11243 pc2dvds 12270 cncnp2m 12984 |
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