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Mirrors > Home > MPE Home > Th. List > vtoclgaf | Structured version Visualization version GIF version |
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Ref | Expression |
---|---|
vtoclgaf.1 | ⊢ Ⅎ𝑥𝐴 |
vtoclgaf.2 | ⊢ Ⅎ𝑥𝜓 |
vtoclgaf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtoclgaf.4 | ⊢ (𝑥 ∈ 𝐵 → 𝜑) |
Ref | Expression |
---|---|
vtoclgaf | ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtoclgaf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | nfel1 2971 | . . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 |
3 | vtoclgaf.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | 2, 3 | nfim 1897 | . . 3 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 → 𝜓) |
5 | eleq1 2877 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
6 | vtoclgaf.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | 5, 6 | imbi12d 348 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))) |
8 | vtoclgaf.4 | . . 3 ⊢ (𝑥 ∈ 𝐵 → 𝜑) | |
9 | 1, 4, 7, 8 | vtoclgf 3513 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ 𝐵 → 𝜓)) |
10 | 9 | pm2.43i 52 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 |
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: ssiun2s 4935 iunopeqop 5376 fvmptss 6757 fvmptf 6766 fmptco 6868 tfis 7549 inar1 10186 sumss 15073 fprodn0 15325 prmind2 16019 lss1d 19728 itg2splitlem 24352 dgrle 24840 cnlnadjlem5 29854 poimirlem25 35082 stoweidlem26 42668 |
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