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Mirrors > Home > MPE Home > Th. List > rspesbca | Structured version Visualization version GIF version |
Description: Existence form of rspsbca 3779. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
rspesbca | ⊢ ((𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ 𝐵 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 3686 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | 1 | rspcev 3527 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑦 ∈ 𝐵 [𝑦 / 𝑥]𝜑) |
3 | cbvrexsvw 3368 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝑦 / 𝑥]𝜑) | |
4 | 2, 3 | sylibr 237 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ 𝐵 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 [wsb 2072 ∈ wcel 2112 ∃wrex 3052 [wsbc 3683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-sbc 3684 |
This theorem is referenced by: spesbc 3781 2nreu 4342 rexopabb 5394 indexfi 8962 indexdom 35578 |
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