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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1468 | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1468.1 | ⊢ (𝜓 → ∀𝑥𝜓) |
bnj1468.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
bnj1468.3 | ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Ref | Expression |
---|---|
bnj1468 | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbccow 3801 | . 2 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑) | |
2 | ax-5 1914 | . . 3 ⊢ (𝜓 → ∀𝑦𝜓) | |
3 | bnj1468.3 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | |
4 | 3 | nfcii 2888 | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 |
5 | 4 | nfeq2 2921 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
6 | nfsbc1v 3798 | . . . . . . 7 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
7 | bnj1468.1 | . . . . . . . 8 ⊢ (𝜓 → ∀𝑥𝜓) | |
8 | 7 | nf5i 2143 | . . . . . . 7 ⊢ Ⅎ𝑥𝜓 |
9 | 6, 8 | nfbi 1907 | . . . . . 6 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
10 | 5, 9 | nfim 1900 | . . . . 5 ⊢ Ⅎ𝑥(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) |
11 | 10 | nf5ri 2189 | . . . 4 ⊢ ((𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) → ∀𝑥(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))) |
12 | ax6ev 1974 | . . . . 5 ⊢ ∃𝑥 𝑥 = 𝑦 | |
13 | eqeq1 2737 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴)) | |
14 | bnj1468.2 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
15 | 13, 14 | syl6bir 254 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝐴 → (𝜑 ↔ 𝜓))) |
16 | sbceq1a 3789 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
17 | 16 | bibi1d 344 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝜓))) |
18 | 15, 17 | sylibd 238 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓))) |
19 | 12, 18 | bnj101 33734 | . . . 4 ⊢ ∃𝑥(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) |
20 | 11, 19 | bnj1131 33798 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ 𝜓)) |
21 | 2, 20 | bnj1464 33855 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜓)) |
22 | 1, 21 | bitr3id 285 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 = wceq 1542 ∈ wcel 2107 [wsbc 3778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-v 3477 df-sbc 3779 |
This theorem is referenced by: bnj1463 34066 |
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