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Mirrors > Home > MPE Home > Th. List > Mathboxes > rspsbc2 | Structured version Visualization version GIF version |
Description: rspsbc 3484 with two quantifying variables. This proof is rspsbc2VD 38112 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rspsbc2 | ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idd 24 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → 𝐶 ∈ 𝐷)) | |
2 | rspsbc 3484 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑)) | |
3 | 2 | a1d 25 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑))) |
4 | sbcralg 3480 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑 ↔ ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑)) | |
5 | 4 | biimpd 218 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑 → ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑)) |
6 | 3, 5 | syl6d 73 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑))) |
7 | rspsbc 3484 | . 2 ⊢ (𝐶 ∈ 𝐷 → (∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)) | |
8 | 1, 6, 7 | syl10 77 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∀wral 2896 [wsbc 3402 |
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-ral 2901 df-v 3175 df-sbc 3403 |
This theorem is referenced by: tratrb 37767 tratrbVD 38119 |
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