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Mirrors > Home > MPE Home > Th. List > rexeqbid | Structured version Visualization version GIF version |
Description: Equality deduction for restricted existential quantifier. (Contributed by Thierry Arnoux, 8-Mar-2017.) |
Ref | Expression |
---|---|
raleqbid.0 | ⊢ Ⅎ𝑥𝜑 |
raleqbid.1 | ⊢ Ⅎ𝑥𝐴 |
raleqbid.2 | ⊢ Ⅎ𝑥𝐵 |
raleqbid.3 | ⊢ (𝜑 → 𝐴 = 𝐵) |
raleqbid.4 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexeqbid | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleqbid.3 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | raleqbid.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | raleqbid.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
4 | 2, 3 | rexeqf 3400 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓)) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓)) |
6 | raleqbid.0 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
7 | raleqbid.4 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
8 | 6, 7 | rexbid 3322 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
9 | 5, 8 | bitrd 281 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 Ⅎwnf 1784 Ⅎwnfc 2963 ∃wrex 3141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rex 3146 |
This theorem is referenced by: iuneq12df 4947 |
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