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| Mirrors > Home > MPE Home > Th. List > rexeqbidva | Structured version Visualization version GIF version | ||
| Description: Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.) |
| Ref | Expression |
|---|---|
| raleqbidva.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| raleqbidva.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| rexeqbidva | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | raleqbidva.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
| 2 | 1 | rexbidva 3177 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
| 3 | raleqbidva.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 4 | 3 | rexeqdv 3327 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜒 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
| 5 | 2, 4 | bitrd 279 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃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: rexeqbidvv 3336 catpropd 17752 addsval 27995 istrkgcb 28464 isperp 28720 perpcom 28721 eengtrkg 29001 eengtrkge 29002 opprqusdrng 33521 fldextrspunlsplem 33723 afsval 34686 matunitlindflem2 37624 rrxlines 48654 |
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