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Mirrors > Home > NFE Home > Th. List > gencl | GIF version |
Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
Ref | Expression |
---|---|
gencl.1 | ⊢ (θ ↔ ∃x(χ ∧ A = B)) |
gencl.2 | ⊢ (A = B → (φ ↔ ψ)) |
gencl.3 | ⊢ (χ → φ) |
Ref | Expression |
---|---|
gencl | ⊢ (θ → ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gencl.1 | . 2 ⊢ (θ ↔ ∃x(χ ∧ A = B)) | |
2 | gencl.3 | . . . . 5 ⊢ (χ → φ) | |
3 | gencl.2 | . . . . 5 ⊢ (A = B → (φ ↔ ψ)) | |
4 | 2, 3 | syl5ib 210 | . . . 4 ⊢ (A = B → (χ → ψ)) |
5 | 4 | impcom 419 | . . 3 ⊢ ((χ ∧ A = B) → ψ) |
6 | 5 | exlimiv 1634 | . 2 ⊢ (∃x(χ ∧ A = B) → ψ) |
7 | 1, 6 | sylbi 187 | 1 ⊢ (θ → ψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 |
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 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 |
This theorem is referenced by: 2gencl 2888 3gencl 2889 |
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