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| 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 3326 | . 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 1540 ∀wral 3061 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ral 3062 df-rex 3071 |
| This theorem is referenced by: subgpgp 19615 znf1o 21570 perfopn 23193 ordtt1 23387 tx1stc 23658 xkococnlem 23667 dgrlem 26268 dchrisum0flb 27554 wlknewwlksn 29907 sigaclcu3 34123 subfacp1lem3 35187 poimirlem1 37628 |
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