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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 2116, ax-ext 2706. (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 3237 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜒)) |
4 | ralnex 3070 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓) | |
5 | ralnex 3070 | . . 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 3059 ∃wrex 3068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-clel 2814 df-ral 3060 df-rex 3069 |
This theorem is referenced by: tfrlem3a 8416 2sqmo 27496 trgcopy 28827 trgcopyeu 28829 acopyeu 28857 tgasa1 28881 dispcmp 33820 satffunlem1lem1 35387 satffunlem2lem1 35389 cbviundavw 36245 f1omptsn 37320 pibt2 37400 prjsprel 42591 opnneilem 48702 |
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