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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcaltop | Structured version Visualization version GIF version |
Description: Distribution of class substitution over alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.) |
Ref | Expression |
---|---|
sbcaltop | ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcsb1v 3914 | . . . 4 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐶 | |
2 | nfcsb1v 3914 | . . . 4 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐷 | |
3 | 1, 2 | nfaltop 35566 | . . 3 ⊢ Ⅎ𝑥⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫ |
4 | 3 | a1i 11 | . 2 ⊢ (𝐴 ∈ V → Ⅎ𝑥⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫) |
5 | csbeq1a 3903 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝐶 = ⦋𝐴 / 𝑥⦌𝐶) | |
6 | altopeq1 35549 | . . . 4 ⊢ (𝐶 = ⦋𝐴 / 𝑥⦌𝐶 → ⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, 𝐷⟫) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝑥 = 𝐴 → ⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, 𝐷⟫) |
8 | csbeq1a 3903 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝐷 = ⦋𝐴 / 𝑥⦌𝐷) | |
9 | altopeq2 35550 | . . . 4 ⊢ (𝐷 = ⦋𝐴 / 𝑥⦌𝐷 → ⟪⦋𝐴 / 𝑥⦌𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝑥 = 𝐴 → ⟪⦋𝐴 / 𝑥⦌𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫) |
11 | 7, 10 | eqtrd 2767 | . 2 ⊢ (𝑥 = 𝐴 → ⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫) |
12 | 4, 11 | csbiegf 3923 | 1 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Ⅎwnfc 2878 Vcvv 3469 ⦋csb 3889 ⟪caltop 35542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-sn 4625 df-pr 4627 df-altop 35544 |
This theorem is referenced by: (None) |
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