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Mirrors > Home > MPE Home > Th. List > cbvralv | Structured version Visualization version GIF version |
Description: Change the bound variable of a restricted universal quantifier using implicit substitution. See cbvralvw 3348 based on fewer axioms , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvralvw 3348 when possible. (Contributed by NM, 28-Jan-1997.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvralv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvralv | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1922 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1922 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvralv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvral 3344 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wral 3051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-10 2143 ax-11 2160 ax-12 2177 ax-13 2371 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2073 df-clel 2809 df-nfc 2879 df-ral 3056 |
This theorem is referenced by: cbvral2v 3364 cbvral3v 3366 |
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