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| Mirrors > Home > MPE Home > Th. List > cbvrexdva | Structured version Visualization version GIF version | ||
| Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) Avoid ax-9 2118, ax-ext 2708. (Revised by Wolf Lammen, 9-Mar-2025.) | 
| Ref | Expression | 
|---|---|
| cbvraldva.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| cbvrexdva | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐴 𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cbvraldva.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
| 2 | 1 | notbid 318 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (¬ 𝜓 ↔ ¬ 𝜒)) | 
| 3 | 2 | cbvraldva 3239 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜒)) | 
| 4 | ralnex 3072 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓) | |
| 5 | ralnex 3072 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝜒 ↔ ¬ ∃𝑦 ∈ 𝐴 𝜒) | |
| 6 | 3, 4, 5 | 3bitr3g 313 | . 2 ⊢ (𝜑 → (¬ ∃𝑥 ∈ 𝐴 𝜓 ↔ ¬ ∃𝑦 ∈ 𝐴 𝜒)) | 
| 7 | 6 | con4bid 317 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐴 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∀wral 3061 ∃wrex 3070 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-clel 2816 df-ral 3062 df-rex 3071 | 
| This theorem is referenced by: tfrlem3a 8417 2sqmo 27481 trgcopy 28812 trgcopyeu 28814 acopyeu 28842 tgasa1 28866 dispcmp 33858 satffunlem1lem1 35407 satffunlem2lem1 35409 cbviundavw 36263 f1omptsn 37338 pibt2 37418 prjsprel 42614 opnneilem 48803 | 
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