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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rexlimdvaacbv | Structured version Visualization version GIF version | ||
| Description: Unpack a restricted existential antecedent while changing the variable with implicit substitution. The equivalent of this theorem without the bound variable change is rexlimdvaa 3156. (Contributed by Rohan Ridenour, 3-Aug-2023.) | 
| Ref | Expression | 
|---|---|
| rexlimdvaacbv.1 | ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) | 
| rexlimdvaacbv.2 | ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜃)) → 𝜒) | 
| Ref | Expression | 
|---|---|
| rexlimdvaacbv | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rexlimdvaacbv.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) | |
| 2 | 1 | cbvrexv 3365 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐴 𝜃) | 
| 3 | rexlimdvaacbv.2 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜃)) → 𝜒) | |
| 4 | 3 | rexlimdvaa 3156 | . 2 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝜃 → 𝜒)) | 
| 5 | 2, 4 | biimtrid 242 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∃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 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 | 
| This theorem is referenced by: rexlimddvcbv 44220 | 
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