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Mirrors > Home > MPE Home > Th. List > 2ralsng | Structured version Visualization version GIF version |
Description: Substitution expressed in terms of two quantifications over singletons. (Contributed by AV, 22-Dec-2019.) |
Ref | Expression |
---|---|
ralsng.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
2ralsng.1 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
2ralsng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralsng.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | 1 | ralbidv 2986 | . . 3 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ {𝐵}𝜑 ↔ ∀𝑦 ∈ {𝐵}𝜓)) |
3 | 2 | ralsng 4218 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑 ↔ ∀𝑦 ∈ {𝐵}𝜓)) |
4 | 2ralsng.1 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
5 | 4 | ralsng 4218 | . 2 ⊢ (𝐵 ∈ 𝑊 → (∀𝑦 ∈ {𝐵}𝜓 ↔ 𝜒)) |
6 | 3, 5 | sylan9bb 736 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {csn 4177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-v 3202 df-sbc 3436 df-sn 4178 |
This theorem is referenced by: mat1ghm 20289 mat1mhm 20290 c0snmgmhm 41914 zrrnghm 41917 |
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