Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 1257 | . . 3 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) |
7 | 4, 6 | sbcied 3439 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ([𝐶 / 𝑧]𝜑 ↔ 𝜓)) |
8 | 1, 2, 7 | sbc2ie 3472 | 1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑 ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 [wsbc 3402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 df-sbc 3403 |
This theorem is referenced by: isdlat 17016 islmod 18690 isslmd 29086 hdmap1fval 36104 hdmapfval 36137 hgmapfval 36196 rmydioph 36599 |
Copyright terms: Public domain | W3C validator |