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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cbvalvv | Structured version Visualization version GIF version |
Description: Version of cbvalv 2261 with a dv condition, which does not require ax-13 2234. UPDATE: this is cbvalvw 1956 (which is proved with fewer axioms). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-cbvalvv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
bj-cbvalvv | ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1830 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | bj-cbvalvv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvalv1 2163 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 |
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 |
This theorem is referenced by: bj-zfpow 31983 bj-nfcjust 32044 |
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