Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
||
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 3402 | . . . 4 ⊢ 𝑦 ∈ V | |
2 | fneq1 6448 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 Fn 𝐴 ↔ 𝑦 Fn 𝐴)) | |
3 | 1, 2 | sbcie 3726 | . . 3 ⊢ ([𝑦 / 𝑥]𝑥 Fn 𝐴 ↔ 𝑦 Fn 𝐴) |
4 | 3 | sbcbii 3743 | . 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝑥 Fn 𝐴 ↔ [𝑧 / 𝑦]𝑦 Fn 𝐴) |
5 | sbccow 3706 | . 2 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝑥 Fn 𝐴 ↔ [𝑧 / 𝑥]𝑥 Fn 𝐴) | |
6 | vex 3402 | . . 3 ⊢ 𝑧 ∈ V | |
7 | fneq1 6448 | . . 3 ⊢ (𝑦 = 𝑧 → (𝑦 Fn 𝐴 ↔ 𝑧 Fn 𝐴)) | |
8 | 6, 7 | sbcie 3726 | . 2 ⊢ ([𝑧 / 𝑦]𝑦 Fn 𝐴 ↔ 𝑧 Fn 𝐴) |
9 | 4, 5, 8 | 3bitr3i 304 | 1 ⊢ ([𝑧 / 𝑥]𝑥 Fn 𝐴 ↔ 𝑧 Fn 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 [wsbc 3683 Fn wfn 6353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-fun 6360 df-fn 6361 |
This theorem is referenced by: bnj156 32373 bnj976 32424 bnj581 32555 |
Copyright terms: Public domain | W3C validator |