![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > raleqtrdv | Structured version Visualization version GIF version |
Description: Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025.) |
Ref | Expression |
---|---|
raleqtrdv.1 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) |
raleqtrdv.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
raleqtrdv | ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleqtrdv.1 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) | |
2 | raleqtrdv.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 2 | raleqdv 3324 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓)) |
4 | 1, 3 | mpbid 232 | 1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∀wral 3059 |
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 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-cleq 2727 df-ral 3060 df-rex 3069 |
This theorem is referenced by: subgpgp 19630 znf1o 21588 perfopn 23209 ordtt1 23403 tx1stc 23674 xkococnlem 23683 dgrlem 26283 dchrisum0flb 27569 wlknewwlksn 29917 sigaclcu3 34103 subfacp1lem3 35167 poimirlem1 37608 |
Copyright terms: Public domain | W3C validator |