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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rspsbc2VD | Structured version Visualization version GIF version | ||
| Description: Virtual deduction proof of rspsbc2 44554.  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 44633 | . . . . 5 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 ▶ 𝐶 ∈ 𝐷 ) | |
| 2 | idn1 44594 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 ▶ 𝐴 ∈ 𝐵 ) | |
| 3 | idn3 44635 | . . . . . . 7 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 ▶ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 ) | |
| 4 | rspsbc 3879 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑)) | |
| 5 | 2, 3, 4 | e13 44768 | . . . . . 6 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 ▶ [𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑 ) | 
| 6 | sbcralg 3874 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑 ↔ ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑)) | |
| 7 | 6 | biimpd 229 | . . . . . 6 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑 → ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑)) | 
| 8 | 2, 5, 7 | e13 44768 | . . . . 5 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 ▶ ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑 ) | 
| 9 | rspsbc 3879 | . . . . 5 ⊢ (𝐶 ∈ 𝐷 → (∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)) | |
| 10 | 1, 8, 9 | e23 44775 | . . . 4 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 , ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 ▶ [𝐶 / 𝑦][𝐴 / 𝑥]𝜑 ) | 
| 11 | 10 | in3 44629 | . . 3 ⊢ ( 𝐴 ∈ 𝐵 , 𝐶 ∈ 𝐷 ▶ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑) ) | 
| 12 | 11 | in2 44625 | . 2 ⊢ ( 𝐴 ∈ 𝐵 ▶ (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)) ) | 
| 13 | 12 | in1 44591 | 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 2108 ∀wral 3061 [wsbc 3788 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-v 3482 df-sbc 3789 df-vd1 44590 df-vd2 44598 df-vd3 44610 | 
| This theorem is referenced by: (None) | 
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