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Mirrors > Home > MPE Home > Th. List > Mathboxes > rspsbc2VD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of rspsbc2 42908. The following user's proof is
completed by invoking mmj2's unify command and using mmj2's StepSelector
to pick all remaining steps of the Metamath proof.
|
Ref | Expression |
---|---|
rspsbc2VD | ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idn2 42987 | . . . . 5 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 ▶ 𝐶 ∈ 𝐷 ) | |
2 | idn1 42948 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 ) | |
3 | idn3 42989 | . . . . . . 7 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 ▶ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 ) | |
4 | rspsbc 3839 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑)) | |
5 | 2, 3, 4 | e13 43122 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 ▶ [𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑 ) |
6 | sbcralg 3834 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑 ↔ ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑)) | |
7 | 6 | biimpd 228 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑 → ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑)) |
8 | 2, 5, 7 | e13 43122 | . . . . 5 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 ▶ ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑 ) |
9 | rspsbc 3839 | . . . . 5 ⊢ (𝐶 ∈ 𝐷 → (∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)) | |
10 | 1, 8, 9 | e23 43129 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 ▶ [𝐶 / 𝑦][𝐴 / 𝑥]𝜑 ) |
11 | 10 | in3 42983 | . . 3 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 ▶ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑) ) |
12 | 11 | in2 42979 | . 2 ⊢ ( 𝐴 ∈ 𝐵 ▶ (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)) ) |
13 | 12 | in1 42945 | 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 2107 ∀wral 3061 [wsbc 3743 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-v 3449 df-sbc 3744 df-vd1 42944 df-vd2 42952 df-vd3 42964 |
This theorem is referenced by: (None) |
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