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Mirrors > Home > NFE Home > Th. List > ifbieq2d | GIF version |
Description: Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
ifbieq2d.1 | ⊢ (φ → (ψ ↔ χ)) |
ifbieq2d.2 | ⊢ (φ → A = B) |
Ref | Expression |
---|---|
ifbieq2d | ⊢ (φ → if(ψ, C, A) = if(χ, C, B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifbieq2d.1 | . . 3 ⊢ (φ → (ψ ↔ χ)) | |
2 | 1 | ifbid 3680 | . 2 ⊢ (φ → if(ψ, C, A) = if(χ, C, A)) |
3 | ifbieq2d.2 | . . 3 ⊢ (φ → A = B) | |
4 | 3 | ifeq2d 3677 | . 2 ⊢ (φ → if(χ, C, A) = if(χ, C, B)) |
5 | 2, 4 | eqtrd 2385 | 1 ⊢ (φ → if(ψ, C, A) = if(χ, C, B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ifcif 3662 |
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-or 359 df-an 360 df-nan 1288 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-rab 2623 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-if 3663 |
This theorem is referenced by: tfineq 4488 |
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