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Mirrors > Home > MPE Home > Th. List > rmobida | Structured version Visualization version GIF version |
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 16-Jun-2017.) |
Ref | Expression |
---|---|
rmobida.1 | ⊢ Ⅎ𝑥𝜑 |
rmobida.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rmobida | ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rmobida.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | rmobida.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | 2 | pm5.32da 579 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))) |
4 | 1, 3 | mobid 2591 | . 2 ⊢ (𝜑 → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))) |
5 | df-rmo 3115 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
6 | df-rmo 3115 | . 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 Ⅎwnf 1769 ∈ wcel 2083 ∃*wmo 2576 ∃*wrmo 3110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-12 2143 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1766 df-nf 1770 df-mo 2578 df-rmo 3115 |
This theorem is referenced by: rmobidva 3355 reuan 3814 |
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