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Mirrors > Home > ILE Home > Th. List > cbvrexdva | 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 2158 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐴) | |
3 | 1, 2 | cbvrexdva2 2688 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐴 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∃wrex 2436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-cleq 2150 df-clel 2153 df-rex 2441 |
This theorem is referenced by: tfrlem3ag 6256 tfrlem3a 6257 tfrlemi1 6279 tfr1onlem3ag 6284 |
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