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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 2955 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | sbcralt 3801 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) | |
3 | 1, 2 | mpan2 690 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2111 Ⅎwnfc 2936 ∀wral 3106 [wsbc 3720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-v 3443 df-sbc 3721 |
This theorem is referenced by: r19.12sn 4616 bnj538 32121 csbwrecsg 34744 cdlemkid3N 38229 cdlemkid4 38230 rspsbc2 41240 rspsbc2VD 41561 |
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