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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 2903 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | sbcrext 3882 | . 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 206 Ⅎwnfc 2888 ∃wrex 3068 [wsbc 3791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-v 3480 df-sbc 3792 |
This theorem is referenced by: 2nreu 4450 ac6sfi 9318 csbwrdg 14579 rexfiuz 15383 2sbcrex 42772 sbc2rex 42775 iccelpart 47358 |
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