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Mathbox for Jonathan Ben-Naim |
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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 3479 | . . . 4 ⊢ 𝑦 ∈ V | |
2 | fneq1 6632 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 Fn 𝐴 ↔ 𝑦 Fn 𝐴)) | |
3 | 1, 2 | sbcie 3818 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 Fn 𝐴 ↔ 𝑦 Fn 𝐴) |
4 | 3 | sbcbii 3835 | . 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝑥 Fn 𝐴 ↔ [𝑧 / 𝑦]𝑦 Fn 𝐴) |
5 | sbccow 3798 | . 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝑥 Fn 𝐴 ↔ [𝑧 / 𝑥]𝑥 Fn 𝐴) | |
6 | vex 3479 | . . 3 ⊢ 𝑧 ∈ V | |
7 | fneq1 6632 | . . 3 ⊢ (𝑦 = 𝑧 → (𝑦 Fn 𝐴 ↔ 𝑧 Fn 𝐴)) | |
8 | 6, 7 | sbcie 3818 | . 2 ⊢ ([𝑧 / 𝑦]𝑦 Fn 𝐴 ↔ 𝑧 Fn 𝐴) |
9 | 4, 5, 8 | 3bitr3i 301 | 1 ⊢ ([𝑧 / 𝑥]𝑥 Fn 𝐴 ↔ 𝑧 Fn 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 [wsbc 3775 Fn wfn 6530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-fun 6537 df-fn 6538 |
This theorem is referenced by: bnj156 33670 bnj976 33719 bnj581 33850 |
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