Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sbc8g | Structured version Visualization version GIF version |
Description: This is the closest we can get to df-sbc 3773 if we start from dfsbcq 3774 (see its comments) and dfsbcq2 3775. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
sbc8g | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq 3774 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | eleq1 2900 | . 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
3 | df-clab 2800 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
4 | equid 2015 | . . . 4 ⊢ 𝑦 = 𝑦 | |
5 | dfsbcq2 3775 | . . . 4 ⊢ (𝑦 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
7 | 3, 6 | bitr2i 278 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 ∈ {𝑥 ∣ 𝜑}) |
8 | 1, 2, 7 | vtoclbg 3569 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 [wsb 2065 ∈ wcel 2110 {cab 2799 [wsbc 3772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-nf 1781 df-clab 2800 df-cleq 2814 df-clel 2893 df-sbc 3773 |
This theorem is referenced by: bnj984 32219 rusbcALT 40764 |
Copyright terms: Public domain | W3C validator |