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Mathbox for Rohan Ridenour |
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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 3154. (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 3363 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐴 𝜃) |
3 | rexlimdvaacbv.2 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝜃)) → 𝜒) | |
4 | 3 | rexlimdvaa 3154 | . 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 2106 ∃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 ax-10 2139 ax-11 2155 ax-12 2175 ax-13 2375 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 |
This theorem is referenced by: rexlimddvcbv 44197 |
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