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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 2590 (and df-cleq 2603 and ax-13 2234) from nfcjust 2739. (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 1830 | . . 3 ⊢ Ⅎ𝑥 𝑦 = 𝑧 | |
2 | bj-eleq1w 32040 | . . 3 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
3 | 1, 2 | nfbidf 2079 | . 2 ⊢ (𝑦 = 𝑧 → (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑧 ∈ 𝐴)) |
4 | 3 | bj-cbvalvv 31920 | 1 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∀wal 1473 Ⅎwnf 1699 ∈ wcel 1977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-nf 1701 df-clel 2606 |
This theorem is referenced by: (None) |
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