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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 rule). (Contributed by NM, 21-Jun-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) |
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 2082 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
4 | abbi 2737 | . 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) ↔ {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) | |
5 | 3, 4 | sylib 208 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 = wceq 1483 Ⅎwnf 1708 {cab 2608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 |
This theorem is referenced by: abbidv 2741 rabeqf 3190 sbcbid 3489 sbceqbidf 29321 opabdm 29423 opabrn 29424 fpwrelmap 29508 bj-rabbida2 32913 iotain 38618 rabbida2 39317 rabbida3 39320 |
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