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Mirrors > Home > MPE Home > Th. List > sbc3ie | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
sbc3ie.1 | ⊢ 𝐴 ∈ V |
sbc3ie.2 | ⊢ 𝐵 ∈ V |
sbc3ie.3 | ⊢ 𝐶 ∈ V |
sbc3ie.4 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbc3ie | ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑 ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbc3ie.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | sbc3ie.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | sbc3ie.3 | . . . 4 ⊢ 𝐶 ∈ V | |
4 | 3 | a1i 11 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 ∈ V) |
5 | sbc3ie.4 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) | |
6 | 5 | 3expa 1153 | . . 3 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) |
7 | 4, 6 | sbcied 3700 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ([𝐶 / 𝑧]𝜑 ↔ 𝜓)) |
8 | 1, 2, 7 | sbc2ie 3731 | 1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑 ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 Vcvv 3415 [wsbc 3663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-12 2222 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-v 3417 df-sbc 3664 |
This theorem is referenced by: isdlat 17547 islmod 19224 isslmd 30301 hdmap1fval 37872 hdmapfval 37903 hgmapfval 37962 rmydioph 38425 |
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