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Mirrors > Home > NFE Home > Th. List > opabbidv | GIF version |
Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 15-May-1995.) |
Ref | Expression |
---|---|
opabbidv.1 | ⊢ (φ → (ψ ↔ χ)) |
Ref | Expression |
---|---|
opabbidv | ⊢ (φ → {〈x, y〉 ∣ ψ} = {〈x, y〉 ∣ χ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1619 | . 2 ⊢ Ⅎxφ | |
2 | nfv 1619 | . 2 ⊢ Ⅎyφ | |
3 | opabbidv.1 | . 2 ⊢ (φ → (ψ ↔ χ)) | |
4 | 1, 2, 3 | opabbid 4624 | 1 ⊢ (φ → {〈x, y〉 ∣ ψ} = {〈x, y〉 ∣ χ}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 {copab 4622 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-opab 4623 |
This theorem is referenced by: opabbii 4626 csbopabg 4637 xpeq1 4798 xpeq2 4799 opabbi2dv 4867 resopab2 5001 cores 5084 f1oiso2 5500 f1od 5726 |
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