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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj538 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) (Proof shortened by OpenAI, 30-Mar-2020.) |
Ref | Expression |
---|---|
bnj538.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
bnj538 | ⊢ ([𝐴 / 𝑦]∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐵 [𝐴 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj538.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | sbcralg 3861 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐵 [𝐴 / 𝑦]𝜑)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ([𝐴 / 𝑦]∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐵 [𝐴 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2098 ∀wral 3053 Vcvv 3466 [wsbc 3770 |
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 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-v 3468 df-sbc 3771 |
This theorem is referenced by: bnj92 34392 bnj539 34421 bnj540 34422 |
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