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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj62 | 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.) |
Ref | Expression |
---|---|
bnj62 | ⊢ ([𝑧 / 𝑥]𝑥 Fn 𝐴 ↔ 𝑧 Fn 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3176 | . . . 4 ⊢ 𝑦 ∈ V | |
2 | fneq1 5893 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 Fn 𝐴 ↔ 𝑦 Fn 𝐴)) | |
3 | 1, 2 | sbcie 3437 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 Fn 𝐴 ↔ 𝑦 Fn 𝐴) |
4 | 3 | sbcbii 3458 | . 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝑥 Fn 𝐴 ↔ [𝑧 / 𝑦]𝑦 Fn 𝐴) |
5 | sbcco 3425 | . 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝑥 Fn 𝐴 ↔ [𝑧 / 𝑥]𝑥 Fn 𝐴) | |
6 | vex 3176 | . . 3 ⊢ 𝑧 ∈ V | |
7 | fneq1 5893 | . . 3 ⊢ (𝑦 = 𝑧 → (𝑦 Fn 𝐴 ↔ 𝑧 Fn 𝐴)) | |
8 | 6, 7 | sbcie 3437 | . 2 ⊢ ([𝑧 / 𝑦]𝑦 Fn 𝐴 ↔ 𝑧 Fn 𝐴) |
9 | 4, 5, 8 | 3bitr3i 289 | 1 ⊢ ([𝑧 / 𝑥]𝑥 Fn 𝐴 ↔ 𝑧 Fn 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 [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-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-fun 5806 df-fn 5807 |
This theorem is referenced by: bnj156 30050 bnj976 30102 bnj581 30232 |
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