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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 582 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))) |
4 | 1, 3 | exbid 2223 | . 2 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))) |
5 | df-rex 3057 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
6 | df-rex 3057 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜒 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)) | |
7 | 4, 5, 6 | 3bitr4g 317 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∃wex 1787 Ⅎwnf 1791 ∈ wcel 2112 ∃wrex 3052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-12 2177 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-nf 1792 df-rex 3057 |
This theorem is referenced by: rexbidvaALT 3228 rexbid 3229 ralrexbidOLD 3232 dfiun2g 4926 iuneq12daf 30569 bnj1366 32476 glbconxN 37078 supminfrnmpt 42599 limsupre2mpt 42889 limsupre3mpt 42893 limsupreuzmpt 42898 |
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