| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > elab2gw | Structured version Visualization version GIF version | ||
| Description: Membership in a class abstraction, using two substitution hypotheses to avoid a disjoint variable condition on 𝑥 and 𝐴, which is not usually significant since 𝐵 is usually a constant. (Contributed by SN, 16-May-2024.) |
| Ref | Expression |
|---|---|
| elabgw.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
| elabgw.2 | ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) |
| elab2gw.3 | ⊢ 𝐵 = {𝑥 ∣ 𝜑} |
| Ref | Expression |
|---|---|
| elab2gw | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ 𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elab2gw.3 | . . 3 ⊢ 𝐵 = {𝑥 ∣ 𝜑} | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
| 3 | elabgw.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 4 | elabgw.2 | . . 3 ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) | |
| 5 | 3, 4 | elabgw 3677 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒)) |
| 6 | 2, 5 | bitrid 283 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 {cab 2714 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 |
| This theorem is referenced by: elrab2w 3696 |
| Copyright terms: Public domain | W3C validator |