Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sbcex2 | Structured version Visualization version GIF version |
Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) (Revised by NM, 18-Aug-2018.) |
Ref | Expression |
---|---|
sbcex2 | ⊢ ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3445 | . 2 ⊢ ([𝐴 / 𝑦]∃𝑥𝜑 → 𝐴 ∈ V) | |
2 | sbcex 3445 | . . 3 ⊢ ([𝐴 / 𝑦]𝜑 → 𝐴 ∈ V) | |
3 | 2 | exlimiv 1858 | . 2 ⊢ (∃𝑥[𝐴 / 𝑦]𝜑 → 𝐴 ∈ V) |
4 | dfsbcq2 3438 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]∃𝑥𝜑 ↔ [𝐴 / 𝑦]∃𝑥𝜑)) | |
5 | dfsbcq2 3438 | . . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑦]𝜑)) | |
6 | 5 | exbidv 1850 | . . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)) |
7 | sbex 2463 | . . 3 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑) | |
8 | 4, 6, 7 | vtoclbg 3267 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)) |
9 | 1, 3, 8 | pm5.21nii 368 | 1 ⊢ ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∃wex 1704 [wsb 1880 ∈ wcel 1990 Vcvv 3200 [wsbc 3435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-v 3202 df-sbc 3436 |
This theorem is referenced by: sbcabel 3517 csbuni 4466 csbxp 5200 csbdm 5318 sbcfung 5912 bnj89 30787 bnj985 31023 csbwrecsg 33173 csboprabg 33176 sbcexf 33918 onfrALTlem5 38757 |
Copyright terms: Public domain | W3C validator |