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Mirrors > Home > MPE Home > Th. List > abbid | Structured version Visualization version GIF version |
Description: Equivalent wff's yield equal class abstractions (deduction form, with nonfreeness hypothesis). (Contributed by NM, 21-Jun-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) Avoid ax-10 2142 and ax-11 2158. (Revised by Wolf Lammen, 6-May-2023.) |
Ref | Expression |
---|---|
abbid.1 | ⊢ Ⅎ𝑥𝜑 |
abbid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
abbid | ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abbid.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | abbid.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | alrimi 2211 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
4 | abbi1 2861 | . 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) | |
5 | 3, 4 | syl 17 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1536 = wceq 1538 Ⅎwnf 1785 {cab 2776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 |
This theorem is referenced by: rabeqf 3428 rabeqiOLD 3430 sbcbid 3773 sbceqbidf 30257 opabdm 30375 opabrn 30376 fpwrelmap 30495 bj-rabbida2 34361 rabbida2 41768 rabbida3 41771 |
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