|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > rexeqtrrdv | Structured version Visualization version GIF version | ||
| Description: Substitution of equal classes into a restricted existential quantifier. (Contributed by Matthew House, 21-Jul-2025.) | 
| Ref | Expression | 
|---|---|
| rexeqtrrdv.1 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) | 
| rexeqtrrdv.2 | ⊢ (𝜑 → 𝐵 = 𝐴) | 
| Ref | Expression | 
|---|---|
| rexeqtrrdv | ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rexeqtrrdv.1 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) | |
| 2 | rexeqtrrdv.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐴) | |
| 3 | 2 | rexeqdv 3327 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜓)) | 
| 4 | 1, 3 | mpbird 257 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∃wrex 3070 | 
| 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-rex 3071 | 
| This theorem is referenced by: zornn0g 10545 ablfacrplem 20085 ablfac2 20109 2ndcctbss 23463 1stcelcls 23469 wwlksnextsurj 29920 primrootsunit1 42098 | 
| Copyright terms: Public domain | W3C validator |