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Mirrors > Home > MPE Home > Th. List > sbcralg | Structured version Visualization version GIF version |
Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
sbcralg | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2899 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | sbcralt 3867 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) | |
3 | 1, 2 | mpan2 689 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2098 Ⅎwnfc 2879 ∀wral 3058 [wsbc 3778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-v 3475 df-sbc 3779 |
This theorem is referenced by: r19.12sn 4729 csbfrecsg 8296 bnj538 34404 cdlemkid3N 40438 cdlemkid4 40439 rspsbc2 44004 rspsbc2VD 44325 |
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