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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 5894 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑦)) | |
3 | fneq2 5894 | . . 3 ⊢ (𝑦 = 𝑌 → (𝑧 Fn 𝑦 ↔ 𝑧 Fn 𝑌)) | |
4 | 2, 3 | sbcie2g 3436 | . 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 195 ∈ wcel 1977 Vcvv 3173 [wsbc 3402 Fn wfn 5799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 df-sbc 3403 df-fn 5807 |
This theorem is referenced by: bnj121 30194 bnj130 30198 bnj207 30205 |
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