New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > rexbii2 | GIF version |
Description: Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.) |
Ref | Expression |
---|---|
rexbii2.1 | ⊢ ((x ∈ A ∧ φ) ↔ (x ∈ B ∧ ψ)) |
Ref | Expression |
---|---|
rexbii2 | ⊢ (∃x ∈ A φ ↔ ∃x ∈ B ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexbii2.1 | . . 3 ⊢ ((x ∈ A ∧ φ) ↔ (x ∈ B ∧ ψ)) | |
2 | 1 | exbii 1582 | . 2 ⊢ (∃x(x ∈ A ∧ φ) ↔ ∃x(x ∈ B ∧ ψ)) |
3 | df-rex 2620 | . 2 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ)) | |
4 | df-rex 2620 | . 2 ⊢ (∃x ∈ B ψ ↔ ∃x(x ∈ B ∧ ψ)) | |
5 | 2, 3, 4 | 3bitr4i 268 | 1 ⊢ (∃x ∈ A φ ↔ ∃x ∈ B ψ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 ∈ wcel 1710 ∃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 |
This theorem depends on definitions: df-bi 177 df-ex 1542 df-rex 2620 |
This theorem is referenced by: rexeqbii 2645 rexbiia 2647 rexrab 3000 rexdifsn 3843 pw1in 4164 |
Copyright terms: Public domain | W3C validator |