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Mirrors > Home > MPE Home > Th. List > sb8 | Structured version Visualization version GIF version |
Description: Substitution of variable in universal quantifier. (Contributed by NM, 16-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) |
Ref | Expression |
---|---|
sb5rf.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
sb8 | ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb5rf.1 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfs1 2353 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 |
3 | sbequ12 2097 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbval 2259 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∀wal 1473 Ⅎwnf 1699 [wsb 1867 |
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 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 |
This theorem is referenced by: sbhb 2426 sbnf2 2427 sb8eu 2491 abv 3179 sb8iota 5775 mo5f 28708 ax11-pm2 32011 bj-nfcf 32112 wl-sb8eut 32538 sbcalf 33087 |
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