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Mirrors > Home > MPE Home > Th. List > vtoclgf | Structured version Visualization version GIF version |
Description: Implicit substitution of a class for a setvar variable, with bound-variable hypotheses in place of disjoint variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.) |
Ref | Expression |
---|---|
vtoclgf.1 | ⊢ Ⅎ𝑥𝐴 |
vtoclgf.2 | ⊢ Ⅎ𝑥𝜓 |
vtoclgf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtoclgf.4 | ⊢ 𝜑 |
Ref | Expression |
---|---|
vtoclgf | ⊢ (𝐴 ∈ 𝑉 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3459 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | vtoclgf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | issetf 3454 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) |
4 | vtoclgf.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
5 | vtoclgf.4 | . . . . 5 ⊢ 𝜑 | |
6 | vtoclgf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | 5, 6 | mpbii 236 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝜓) |
8 | 4, 7 | exlimi 2215 | . . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓) |
9 | 3, 8 | sylbi 220 | . 2 ⊢ (𝐴 ∈ V → 𝜓) |
10 | 1, 9 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∃wex 1781 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 Vcvv 3441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 |
This theorem is referenced by: vtocl2gf 3518 vtocl3gf 3519 vtoclgaf 3521 elabgf 3610 ssiun2sf 30323 subtr 33775 subtr2 33776 supxrgere 41965 supxrgelem 41969 supxrge 41970 fsumsplit1 42214 fmuldfeqlem1 42224 fprodcnlem 42241 climsuse 42250 dvnmptdivc 42580 dvmptfprodlem 42586 stoweidlem59 42701 fourierdlem31 42780 sge0f1o 43021 sge0fodjrnlem 43055 |
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