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Mirrors > Home > MPE Home > Th. List > sbcrex | Structured version Visualization version GIF version |
Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.) |
Ref | Expression |
---|---|
sbcrex | ⊢ ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2904 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | sbcrext 3817 | . 2 ⊢ (Ⅎ𝑦𝐴 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 Ⅎwnfc 2884 ∃wrex 3070 [wsbc 3727 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ral 3062 df-rex 3071 df-v 3443 df-sbc 3728 |
This theorem is referenced by: 2nreu 4388 ac6sfi 9152 csbwrdg 14347 rexfiuz 15158 2sbcrex 40876 sbc2rex 40879 iccelpart 45244 |
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