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Mirrors > Home > MPE Home > Th. List > rexeqfOLD | Structured version Visualization version GIF version |
Description: Obsolete version of rexeqf 3344 as of 9-Mar-2025. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
raleqf.1 | ⊢ Ⅎ𝑥𝐴 |
raleqf.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
rexeqfOLD | ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleqf.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | raleqf.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | nfeq 2910 | . . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
4 | eleq2 2816 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
5 | 4 | anbi1d 629 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))) |
6 | 3, 5 | exbid 2208 | . 2 ⊢ (𝐴 = 𝐵 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))) |
7 | df-rex 3065 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
8 | df-rex 3065 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
9 | 6, 7, 8 | 3bitr4g 314 | 1 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 Ⅎwnfc 2877 ∃wrex 3064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-cleq 2718 df-clel 2804 df-nfc 2879 df-rex 3065 |
This theorem is referenced by: (None) |
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