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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 3116 | . 2 ⊢ ∀𝑥 ∈ 𝐵 𝜑 |
3 | rspsbc 3808 | . 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 2111 ∀wral 3106 [wsbc 3720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-sbc 3721 |
This theorem is referenced by: (None) |
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