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| Description: Existence form of rspsbca 3879. (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 3790 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
| 2 | 1 | rspcev 3621 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑦 ∈ 𝐵 [𝑦 / 𝑥]𝜑) | 
| 3 | cbvrexsvw 3317 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝑦 / 𝑥]𝜑) | |
| 4 | 2, 3 | sylibr 234 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ 𝐵 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 [wsb 2063 ∈ wcel 2107 ∃wrex 3069 [wsbc 3787 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-sbc 3788 | 
| This theorem is referenced by: spesbc 3881 2nreu 4443 rexopabb 5532 indexfi 9401 indexdom 37742 | 
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