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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj90 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj90.1 | ⊢ 𝑌 ∈ V |
Ref | Expression |
---|---|
bnj90 | ⊢ ([𝑌 / 𝑥]𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj90.1 | . 2 ⊢ 𝑌 ∈ V | |
2 | fneq2 6660 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑦)) | |
3 | fneq2 6660 | . . 3 ⊢ (𝑦 = 𝑌 → (𝑧 Fn 𝑦 ↔ 𝑧 Fn 𝑌)) | |
4 | 2, 3 | sbcie2g 3833 | . 2 ⊢ (𝑌 ∈ V → ([𝑌 / 𝑥]𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑌)) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ ([𝑌 / 𝑥]𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∈ wcel 2105 Vcvv 3477 [wsbc 3790 Fn wfn 6557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-sbc 3791 df-fn 6565 |
This theorem is referenced by: bnj121 34862 bnj130 34866 bnj207 34873 |
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