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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfopdALT | Structured version Visualization version GIF version |
Description: Deduction version of bound-variable hypothesis builder nfop 4687. This shows how the deduction version of a not-free theorem such as nfop 4687 can be created from the corresponding not-free inference theorem. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfopdALT.1 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfopdALT.2 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
Ref | Expression |
---|---|
nfopdALT | ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfopdALT.1 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
2 | nfopdALT.2 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
3 | abidnf 3602 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | |
4 | 3 | adantr 473 | . . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) |
5 | abidnf 3602 | . . . 4 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) | |
6 | 5 | adantl 474 | . . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) |
7 | 4, 6 | opeq12d 4679 | . 2 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 = 〈𝐴, 𝐵〉) |
8 | nfaba1 2933 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} | |
9 | nfaba1 2933 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} | |
10 | 8, 9 | nfop 4687 | . 2 ⊢ Ⅎ𝑥〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 |
11 | 1, 2, 7, 10 | nfded2 35497 | 1 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∀wal 1505 = wceq 1507 ∈ wcel 2048 {cab 2753 Ⅎwnfc 2910 〈cop 4441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-rab 3091 df-v 3411 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 |
This theorem is referenced by: (None) |
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