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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 3856 | . 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 208 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 [wsbc 3771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-v 3496 df-sbc 3772 |
This theorem is referenced by: bnj92 32129 bnj539 32158 bnj540 32159 |
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