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Mirrors > Home > NFE Home > Th. List > rexab | GIF version |
Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
ralab.1 | ⊢ (y = x → (φ ↔ ψ)) |
Ref | Expression |
---|---|
rexab | ⊢ (∃x ∈ {y ∣ φ}χ ↔ ∃x(ψ ∧ χ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2620 | . 2 ⊢ (∃x ∈ {y ∣ φ}χ ↔ ∃x(x ∈ {y ∣ φ} ∧ χ)) | |
2 | vex 2862 | . . . . 5 ⊢ x ∈ V | |
3 | ralab.1 | . . . . 5 ⊢ (y = x → (φ ↔ ψ)) | |
4 | 2, 3 | elab 2985 | . . . 4 ⊢ (x ∈ {y ∣ φ} ↔ ψ) |
5 | 4 | anbi1i 676 | . . 3 ⊢ ((x ∈ {y ∣ φ} ∧ χ) ↔ (ψ ∧ χ)) |
6 | 5 | exbii 1582 | . 2 ⊢ (∃x(x ∈ {y ∣ φ} ∧ χ) ↔ ∃x(ψ ∧ χ)) |
7 | 1, 6 | bitri 240 | 1 ⊢ (∃x ∈ {y ∣ φ}χ ↔ ∃x(ψ ∧ χ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∃wex 1541 ∈ wcel 1710 {cab 2339 ∃wrex 2615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-rex 2620 df-v 2861 |
This theorem is referenced by: opeq 4619 frds 5935 |
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