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Mirrors > Home > MPE Home > Th. List > rexbida | Structured version Visualization version GIF version |
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 6-Oct-2003.) |
Ref | Expression |
---|---|
rexbida.1 | ⊢ Ⅎ𝑥𝜑 |
rexbida.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexbida | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexbida.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | rexbida.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | 2 | pm5.32da 579 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))) |
4 | 1, 3 | exbid 2215 | . 2 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))) |
5 | df-rex 3141 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
6 | df-rex 3141 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜒 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)) | |
7 | 4, 5, 6 | 3bitr4g 315 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∃wex 1771 Ⅎwnf 1775 ∈ wcel 2105 ∃wrex 3136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 df-rex 3141 |
This theorem is referenced by: rexbidvaALT 3316 rexbid 3317 ralrexbidOLD 3320 dfiun2g 4946 dfiun2gOLD 4947 iuneq12daf 30236 bnj1366 32000 glbconxN 36394 supminfrnmpt 41595 limsupre2mpt 41887 limsupre3mpt 41891 limsupreuzmpt 41896 |
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