Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3197 | . . 3 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ {𝐵}𝜑 ↔ ∀𝑦 ∈ {𝐵}𝜓)) |
3 | 2 | ralsng 4606 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑 ↔ ∀𝑦 ∈ {𝐵}𝜓)) |
4 | 2ralsng.1 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
5 | 4 | ralsng 4606 | . 2 ⊢ (𝐵 ∈ 𝑊 → (∀𝑦 ∈ {𝐵}𝜓 ↔ 𝜒)) |
6 | 3, 5 | sylan9bb 512 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐵}𝜑 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {csn 4560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-v 3496 df-sbc 3772 df-sn 4561 |
This theorem is referenced by: mat1ghm 21086 mat1mhm 21087 f1resfz0f1d 32356 c0snmgmhm 44179 zrrnghm 44182 |
Copyright terms: Public domain | W3C validator |