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Mirrors > Home > MPE Home > Th. List > rexanid | Structured version Visualization version GIF version |
Description: Cancellation law for restricted existential quantification. (Contributed by Peter Mazsa, 24-May-2018.) (Proof shortened by Wolf Lammen, 8-Jul-2023.) |
Ref | Expression |
---|---|
rexanid | ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ibar 532 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
2 | 1 | bicomd 226 | . 2 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝜑)) |
3 | 2 | rexbiia 3169 | 1 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2110 ∃wrex 3062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-rex 3067 |
This theorem is referenced by: sn-axrep5v 39907 |
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