Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > currysetlem | Structured version Visualization version GIF version |
Description: Lemma for currysetlem 34757, where it is used with (𝑥 ∈ 𝑥 → 𝜑) substituted for 𝜓. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
currysetlem | ⊢ ({𝑥 ∣ 𝜓} ∈ 𝑉 → ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfab1 2901 | . 2 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓} | |
2 | 1, 1 | nfel 2913 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} |
3 | nfv 1921 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
4 | 2, 3 | nfim 1903 | . 2 ⊢ Ⅎ𝑥({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 𝜑) |
5 | id 22 | . . . 4 ⊢ (𝑥 = {𝑥 ∣ 𝜓} → 𝑥 = {𝑥 ∣ 𝜓}) | |
6 | 5, 5 | eleq12d 2827 | . . 3 ⊢ (𝑥 = {𝑥 ∣ 𝜓} → (𝑥 ∈ 𝑥 ↔ {𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓})) |
7 | 6 | imbi1d 345 | . 2 ⊢ (𝑥 = {𝑥 ∣ 𝜓} → ((𝑥 ∈ 𝑥 → 𝜑) ↔ ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 𝜑))) |
8 | 1, 4, 7 | elabgf 3568 | 1 ⊢ ({𝑥 ∣ 𝜓} ∈ 𝑉 → ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ ({𝑥 ∣ 𝜓} ∈ {𝑥 ∣ 𝜓} → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1542 ∈ wcel 2114 {cab 2716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-v 3400 |
This theorem is referenced by: curryset 34758 |
Copyright terms: Public domain | W3C validator |