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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.) |
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 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
2 | eqidd 2611 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐴) | |
3 | 1, 2 | cbvrexdva2 3152 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∃wrex 2897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-nf 1701 df-cleq 2603 df-clel 2606 df-rex 2902 |
This theorem is referenced by: tfrlem3a 7360 trgcopy 25496 trgcopyeu 25498 acopyeu 25525 tgasa1 25539 2sqmo 28980 dispcmp 29254 f1omptsn 32360 |
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