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Mirrors > Home > MPE Home > Th. List > sbc2ie | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Gino Giotto, 12-Oct-2024.) |
Ref | Expression |
---|---|
sbc2ie.1 | ⊢ 𝐴 ∈ V |
sbc2ie.2 | ⊢ 𝐵 ∈ V |
sbc2ie.3 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbc2ie | ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbc2ie.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | sbc2ie.2 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑥 = 𝐴 → 𝐵 ∈ V) |
4 | sbc2ie.3 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | sbcied 3823 | . 2 ⊢ (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑 ↔ 𝜓)) |
6 | 1, 5 | sbcie 3821 | 1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 [wsbc 3778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-sbc 3779 |
This theorem is referenced by: sbc3ie 3864 brfi1uzind 14459 opfi1uzind 14462 wrd2ind 14673 isprs 18250 isdrs 18254 istos 18371 issrg 20011 isslmd 32347 rexrabdioph 41532 rmydioph 41753 rmxdioph 41755 expdiophlem2 41761 2reu8i 45821 |
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