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Mirrors > Home > MPE Home > Th. List > Mathboxes > rspsbc2VD | Structured version Visualization version GIF version |
Description: Virtual deduction proof of rspsbc2 40359. 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 40438 | . . . . 5 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 ▶ 𝐶 ∈ 𝐷 ) | |
2 | idn1 40399 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 ) | |
3 | idn3 40440 | . . . . . . 7 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 ▶ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 ) | |
4 | rspsbc 3785 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑)) | |
5 | 2, 3, 4 | e13 40573 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 ▶ [𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑 ) |
6 | sbcralg 3780 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑 ↔ ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑)) | |
7 | 6 | biimpd 230 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑 → ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑)) |
8 | 2, 5, 7 | e13 40573 | . . . . 5 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 ▶ ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑 ) |
9 | rspsbc 3785 | . . . . 5 ⊢ (𝐶 ∈ 𝐷 → (∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)) | |
10 | 1, 8, 9 | e23 40580 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 ▶ [𝐶 / 𝑦][𝐴 / 𝑥]𝜑 ) |
11 | 10 | in3 40434 | . . 3 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 ▶ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑) ) |
12 | 11 | in2 40430 | . 2 ⊢ ( 𝐴 ∈ 𝐵 ▶ (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)) ) |
13 | 12 | in1 40396 | 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 2079 ∀wral 3103 [wsbc 3701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ral 3108 df-v 3434 df-sbc 3702 df-vd1 40395 df-vd2 40403 df-vd3 40415 |
This theorem is referenced by: (None) |
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