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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 3175 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
3 | raleqbidva.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | rexeqdv 3325 | . 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 1537 ∈ wcel 2106 ∃wrex 3068 |
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-rex 3069 |
This theorem is referenced by: rexeqbidvv 3334 catpropd 17754 addsval 28010 istrkgcb 28479 isperp 28735 perpcom 28736 eengtrkg 29016 eengtrkge 29017 opprqusdrng 33501 afsval 34665 matunitlindflem2 37604 rrxlines 48583 |
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