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Mirrors > Home > MPE Home > Th. List > sbc2ieOLD | Structured version Visualization version GIF version |
Description: Obsolete version of sbc2ie 3778 as of 12-Oct-2024. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbc2ieOLD.1 | ⊢ 𝐴 ∈ V |
sbc2ieOLD.2 | ⊢ 𝐵 ∈ V |
sbc2ieOLD.3 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbc2ieOLD | ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbc2ieOLD.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | sbc2ieOLD.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | nfv 1922 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
4 | nfv 1922 | . . 3 ⊢ Ⅎ𝑦𝜓 | |
5 | 2 | nfth 1809 | . . 3 ⊢ Ⅎ𝑥 𝐵 ∈ V |
6 | sbc2ieOLD.3 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
7 | 3, 4, 5, 6 | sbc2iegf 3777 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)) |
8 | 1, 2, 7 | mp2an 692 | 1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 [wsbc 3694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-sbc 3695 |
This theorem is referenced by: (None) |
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