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Mirrors > Home > NFE Home > Th. List > abeq2d | GIF version |
Description: Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.) |
Ref | Expression |
---|---|
abeqd.1 | ⊢ (φ → A = {x ∣ ψ}) |
Ref | Expression |
---|---|
abeq2d | ⊢ (φ → (x ∈ A ↔ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abeqd.1 | . . 3 ⊢ (φ → A = {x ∣ ψ}) | |
2 | 1 | eleq2d 2420 | . 2 ⊢ (φ → (x ∈ A ↔ x ∈ {x ∣ ψ})) |
3 | abid 2341 | . 2 ⊢ (x ∈ {x ∣ ψ} ↔ ψ) | |
4 | 2, 3 | syl6bb 252 | 1 ⊢ (φ → (x ∈ A ↔ ψ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ∈ wcel 1710 {cab 2339 |
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-11 1746 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 |
This theorem is referenced by: fvelimab 5370 |
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