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Mirrors > Home > MPE Home > Th. List > sbcth2 | Structured version Visualization version GIF version |
Description: A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
sbcth2.1 | ⊢ (𝑥 ∈ 𝐵 → 𝜑) |
Ref | Expression |
---|---|
sbcth2 | ⊢ (𝐴 ∈ 𝐵 → [𝐴 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcth2.1 | . . 3 ⊢ (𝑥 ∈ 𝐵 → 𝜑) | |
2 | 1 | rgen 3148 | . 2 ⊢ ∀𝑥 ∈ 𝐵 𝜑 |
3 | rspsbc 3862 | . 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → [𝐴 / 𝑥]𝜑)) | |
4 | 2, 3 | mpi 20 | 1 ⊢ (𝐴 ∈ 𝐵 → [𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∀wral 3138 [wsbc 3772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-sbc 3773 |
This theorem is referenced by: (None) |
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