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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 3131 | . 2 ⊢ ∀𝑥 ∈ 𝐵 𝜑 |
3 | rspsbc 3742 | . 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 2164 ∀wral 3117 [wsbc 3662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-v 3416 df-sbc 3663 |
This theorem is referenced by: (None) |
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