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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbcrexgOLD | Structured version Visualization version GIF version |
Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcrex 3804 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
sbcrexgOLD | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 3723 | . 2 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑)) | |
2 | dfsbcq2 3723 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | 2 | rexbidv 3256 | . 2 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
4 | nfcv 2955 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
5 | nfs1v 2157 | . . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
6 | 4, 5 | nfrex 3268 | . . 3 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 |
7 | sbequ12 2250 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
8 | 7 | rexbidv 3256 | . . 3 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)) |
9 | 6, 8 | sbie 2521 | . 2 ⊢ ([𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑) |
10 | 1, 3, 9 | vtoclbg 3517 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 [wsb 2069 ∈ wcel 2111 ∃wrex 3107 [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-13 2379 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 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-rex 3112 df-sbc 3721 |
This theorem is referenced by: 2sbcrexOLD 39727 sbc2rexgOLD 39729 |
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