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Mirrors > Home > MPE Home > Th. List > nfceqdf | Structured version Visualization version GIF version |
Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
nfceqdf.1 | ⊢ Ⅎ𝑥𝜑 |
nfceqdf.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
nfceqdf | ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfceqdf.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | nfceqdf.2 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 2 | eleq2d 2875 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
4 | 1, 3 | nfbidf 2224 | . . 3 ⊢ (𝜑 → (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐵)) |
5 | 4 | albidv 1921 | . 2 ⊢ (𝜑 → (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)) |
6 | df-nfc 2938 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) | |
7 | df-nfc 2938 | . 2 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵) | |
8 | 5, 6, 7 | 3bitr4g 317 | 1 ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1536 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 |
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-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 df-cleq 2791 df-clel 2870 df-nfc 2938 |
This theorem is referenced by: nfopd 4782 dfnfc2 4822 nfimad 5905 nffvd 6657 riotasv2d 36253 nfcxfrdf 36262 nfded 36263 nfded2 36264 nfunidALT2 36265 |
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