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Mirrors > Home > MPE Home > Th. List > cbvralv2 | Structured version Visualization version GIF version |
Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvralv2.1 | ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) |
cbvralv2.2 | ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
cbvralv2 | ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2899 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2899 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | nfv 1921 | . 2 ⊢ Ⅎ𝑦𝜓 | |
4 | nfv 1921 | . 2 ⊢ Ⅎ𝑥𝜒 | |
5 | cbvralv2.2 | . 2 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
6 | cbvralv2.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | |
7 | 1, 2, 3, 4, 5, 6 | cbvralcsf 3832 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1542 ∀wral 3053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-13 2372 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-sbc 3681 df-csb 3791 |
This theorem is referenced by: (None) |
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