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Mirrors > Home > MPE Home > Th. List > abeq2fOLD | Structured version Visualization version GIF version |
Description: Obsolete version of abeq2f 3013 as of 13-May-2023. (Contributed by Thierry Arnoux, 11-May-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
abeq2f.0 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
abeq2fOLD | ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abeq2f.0 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | nfcrii 2970 | . . 3 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
3 | hbab1 2807 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
4 | 2, 3 | cleqh 2936 | . 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑})) |
5 | abid 2803 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
6 | 5 | bibi2i 340 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑥 ∈ 𝐴 ↔ 𝜑)) |
7 | 6 | albii 1820 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) |
8 | 4, 7 | bitri 277 | 1 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1535 = wceq 1537 ∈ wcel 2114 {cab 2799 Ⅎwnfc 2961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 |
This theorem is referenced by: (None) |
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