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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nfcjust | Structured version Visualization version GIF version |
Description: Remove dependency on ax-ext 2602 (and df-cleq 2615 and ax-13 2246) from nfcjust 2752. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nfcjust | ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1843 | . . 3 ⊢ Ⅎ𝑥 𝑦 = 𝑧 | |
2 | eleq1w 2684 | . . 3 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
3 | 1, 2 | nfbidf 2092 | . 2 ⊢ (𝑦 = 𝑧 → (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑧 ∈ 𝐴)) |
4 | 3 | bj-cbvalvv 32733 | 1 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∀wal 1481 Ⅎwnf 1708 ∈ wcel 1990 |
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-10 2019 ax-11 2034 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-nf 1710 df-clel 2618 |
This theorem is referenced by: (None) |
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